Initial Setup: The Polynomial Expression

We are tasked with factorising the following algebraic expression:

$P(a, b) = 8a^3 - b^3 - 12a^2b + 6ab^2$

Step 1: Identifying the Relevant Algebraic Identity

By observing the degree and the signs of the terms in the polynomial, we note that it consists of four terms: two perfect cubes ($8a^3$ and $-b^3$) and two cross-terms. This structure strongly suggests the expansion of the cube of a binomial difference. [Per the standard algebraic identities for polynomials], the relevant formula is:

$(x - y)^3 = x^3 - y^3 - 3x^2y + 3xy^2$

Step 2: Structural Mapping and Term Transformation

To apply the identity, we must express each term of the given polynomial in the exact form of the identity's expansion. We will determine the base values for $x$ and $y$ by taking the cube roots of the perfect cube terms.

• First term ($x^3$): The term $8a^3$ can be rewritten as a perfect cube. Since $2^3 = 8$, we have $8a^3 = (2a)^3$. Thus, we set $x = 2a$.
• Second term ($-y^3$): The term $-b^3$ can be rewritten as $-(b)^3$. Thus, we set $y = b$.

Now, we must verify if the remaining terms ($-12a^2b$ and $6ab^2$) perfectly match the $-3x^2y$ and $+3xy^2$ components of the identity using our established values for $x$ and $y$.

• Third term ($-3x^2y$): Substituting $x = 2a$ and $y = b$:
  $-3(2a)^2(b) = -3(4a^2)(b) = -12a^2b$.
  [This perfectly matches the third term of our given polynomial.]
• Fourth term ($+3xy^2$): Substituting $x = 2a$ and $y = b$:
  $+3(2a)(b)^2 = 6ab^2$.
  [This perfectly matches the fourth term of our given polynomial.]

Visualizing the Algebraic Mapping

The following diagram illustrates the structural equivalence between the given polynomial and the standard algebraic identity.

Step 3: Synthesizing the Factored Form

Since all terms of the polynomial $8a^3 - b^3 - 12a^2b + 6ab^2$ map flawlessly to the expansion of $(x - y)^3$ where $x = 2a$ and $y = b$, we can condense the expanded polynomial back into its factored binomial form.

$8a^3 - b^3 - 12a^2b + 6ab^2 = (2a)^3 - (b)^3 - 3(2a)^2(b) + 3(2a)(b)^2$

$= (2a - b)^3$

Step 4: Expanding into Linear Factors

Factorisation requires expressing the polynomial as a product of its irreducible linear factors. The exponent $3$ indicates that the binomial $(2a - b)$ is multiplied by itself three times.

$(2a - b)^3 = (2a - b)(2a - b)(2a - b)$

Final Solution: The completely factorised form of the polynomial $8a^3 - b^3 - 12a^2b + 6ab^2$ is $(2a - b)(2a - b)(2a - b)$.

Given Expression & Algebraic Identity

We are tasked with expanding the following trinomial squared:

$ \left( \frac{1}{4}a - \frac{1}{2}b + 1 \right)^2 $

To expand this expression systematically, we utilize the standard algebraic identity for the square of a trinomial [Derived from the distributive property of multiplication over addition]:

$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx $

Step 1: Variable Mapping

By comparing our given expression $\left( \frac{1}{4}a - \frac{1}{2}b + 1 \right)^2$ with the standard identity $(x + y + z)^2$, we can establish a direct mapping of the terms. It is critical to include the negative sign in our mapping to maintain algebraic integrity.

• $ x = \frac{1}{4}a $
• $ y = -\frac{1}{2}b $
• $ z = 1 $

Step 2: Substitution into the Identity

Substituting the mapped variables into the expanded form of the identity, we get:

$ \left( \frac{1}{4}a \right)^2 + \left( -\frac{1}{2}b \right)^2 + (1)^2 + 2\left( \frac{1}{4}a \right)\left( -\frac{1}{2}b \right) + 2\left( -\frac{1}{2}b \right)(1) + 2(1)\left( \frac{1}{4}a \right) $

Step 3: Term-by-Term Simplification

We will now apply the exponent rules [specifically $(uv)^n = u^n v^n$] and perform scalar multiplication for each distinct term.

Step 4: Final Assembly

Combining all the simplified terms from Step 3 yields the fully expanded polynomial. We write the terms in descending order of degree where applicable, though standard expansion order is perfectly rigorous:

$ \frac{1}{16}a^2 + \frac{1}{4}b^2 + 1 - \frac{1}{4}ab - b + \frac{1}{2}a $

Final Solution: The expanded form of the given polynomial is $ \frac{1}{16}a^2 + \frac{1}{4}b^2 + 1 - \frac{1}{4}ab - b + \frac{1}{2}a $

Step 1: Initial Setup & Identification of the Algebraic Identity

We are tasked with finding the product of the given binomials:

$ \left(y^2 + \frac{3}{2}\right) \left(y^2 - \frac{3}{2}\right) $

By analyzing the structural form of the expression, we observe that it consists of the product of the sum and difference of the exact same two terms. This perfectly matches the fundamental algebraic identity for the Difference of Squares:

$ (a + b)(a - b) = a^2 - b^2 $

[Theoretical Justification: The cross-terms in the expansion $(a)(a) - (a)(b) + (b)(a) - (b)(b)$ cancel out, leaving only the squared terms $a^2 - b^2$.]

Step 2: Variable Mapping and Substitution

To apply the identity rigorously, we establish a one-to-one correspondence between the variables in the identity and the terms in our specific expression.

• Let the first term $a = y^2$
• Let the second term $b = \frac{3}{2}$

Substituting these defined values into the right-hand side of the Difference of Squares identity ($a^2 - b^2$), we formulate the following equation:

$ \left(y^2\right)^2 - \left(\frac{3}{2}\right)^2 $

Step 3: Geometric Verification of the Identity (Visual Proof)

The algebraic identity $(a-b)(a+b) = a^2 - b^2$ can be geometrically proven by analyzing the area of a square of side $a$ with a smaller square of side $b$ removed. The remaining area can be rearranged into a rectangle with dimensions $(a+b)$ and $(a-b)$.

Step 4: Execution of the Algebraic Expansion & Simplification

We now simplify the expression $\left(y^2\right)^2 - \left(\frac{3}{2}\right)^2$ by applying the fundamental laws of exponents.

1. Simplifying the first term $\left(y^2\right)^2$:

According to the Power of a Power Property, $(x^m)^n = x^{m \cdot n}$. Therefore, we multiply the exponents:

$ \left(y^2\right)^2 = y^{2 \times 2} = y^4 $

2. Simplifying the second term $\left(\frac{3}{2}\right)^2$:

According to the Power of a Quotient Property, $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$. We distribute the square to both the numerator and the denominator:

$ \left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} $

Step 5: Final Polynomial Construction

By combining the simplified terms from Step 4 back into our expanded equation structure, we arrive at the final evaluated polynomial.

$ y^4 - \frac{9}{4} $

Final Solution: The product of the given binomials is $y^4 - \frac{9}{4}$.

Initial Setup & Structural Analysis

We are tasked with factorising the following algebraic expression of six terms:

$P(x, y, z) = 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz$

[Per the fundamental theorems of polynomial algebra], an expression containing three perfect square terms and three cross-product terms strongly indicates the expansion of a squared trinomial. The governing algebraic identity is:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Step 1: Extraction of Base Variables (Magnitude)

We must map the first three terms of our polynomial to the squared terms of the identity ($a^2, b^2, c^2$) to find the magnitudes of $a, b,$ and $c$.

• First term: $a^2 = 4x^2 \implies |a| = \sqrt{4x^2} = 2x$
• Second term: $b^2 = 9y^2 \implies |b| = \sqrt{9y^2} = 3y$
• Third term: $c^2 = 16z^2 \implies |c| = \sqrt{16z^2} = 4z$

Step 2: Sign Determination via Cross-Product Analysis

The signs of $a, b,$ and $c$ are determined by analyzing the signs of the cross-product terms ($2ab, 2bc, 2ca$).

The given cross-product terms are:

• $2ab = +12xy$ (Positive)
• $2bc = -24yz$ (Negative)
• $2ca = -16xz$ (Negative)

[By the rules of integer multiplication], the product $2ab$ is positive, which dictates that $a$ and $b$ must share the same sign. Let us assume both $a$ and $b$ are positive:

$a = +2x$

$b = +3y$

The products $2bc$ and $2ca$ are both negative. Since $b$ is positive, for $2bc$ to be negative, $c$ must be negative. Similarly, since $a$ is positive, for $2ca$ to be negative, $c$ must be negative. This confirms that the negative sign originates exclusively from the $z$-term.

$c = -4z$

Step 3: Verification of the Identity

We substitute $a = 2x$, $b = 3y$, and $c = -4z$ back into the expanded identity to ensure absolute mathematical equivalence:

$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
$= (2x)^2 + (3y)^2 + (-4z)^2 + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x)$
$= 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz$

The expanded form perfectly matches the original polynomial. Therefore, the expression can be written as the square of the trinomial $(2x + 3y - 4z)$.

Visualizing the Expansion (Tabular Area Model)

To rigorously prove the distribution of terms, we can use a tabular area model. The sum of all cells in the $3 \times 3$ matrix equals the original polynomial, demonstrating how the cross-terms combine.

Notice how the symmetric off-diagonal terms combine perfectly: $(6xy + 6xy = 12xy)$, $(-12yz - 12yz = -24yz)$, and $(-8xz - 8xz = -16xz)$.

Step 4: Final Synthesis

Having established the base terms and their respective signs, we write the expression in its fully factorised form. Since factorisation requires expressing the polynomial as a product of its irreducible factors, we write the squared binomial as the product of two identical brackets.

Final Solution: The factorised form of the given polynomial is $(2x + 3y - 4z)(2x + 3y - 4z)$, which can also be written as $(2x + 3y - 4z)^2$.

Initial Setup & Given Polynomial

We are tasked with factorising the following algebraic expression:

$P(a) = 27 - 125a^3 - 135a + 225a^2$

Step 1: Identifying the Relevant Algebraic Identity

By analyzing the polynomial, we observe the presence of perfect cubes ($27$ and $125a^3$) and alternating signs. This structural pattern strongly indicates the application of the standard cubic identity for the difference of two terms [Per the Binomial Theorem for exponent 3]:

$(x - y)^3 = x^3 - y^3 - 3x^2y + 3xy^2$

To utilize this identity, we must map the terms of our given polynomial to the terms of the expansion.

Step 2: Extracting the Base Terms (Cube Roots)

We isolate the perfect cube terms to determine our candidate values for $x$ and $y$.

• First term (Constant): The term $27$ is a perfect cube.
  $27 = (3)^3 \implies x = 3$
• Second term (Cubic): The term $125a^3$ is a perfect cube.
  $125a^3 = (5a)^3 \implies y = 5a$

Step 3: Verifying the Cross Terms

To rigorously confirm that the polynomial fits the identity $(x - y)^3$, we must evaluate the intermediate cross terms $-3x^2y$ and $+3xy^2$ using our candidate values $x = 3$ and $y = 5a$.

• Evaluating $-3x^2y$:
  $-3(3)^2(5a) = -3(9)(5a) = -27(5a) = -135a$
  [This perfectly matches the third term of the given polynomial].
• Evaluating $+3xy^2$:
  $+3(3)(5a)^2 = 9(25a^2) = 225a^2$
  [This perfectly matches the fourth term of the given polynomial].

Visual Mapping of the Algebraic Identity

The following diagram illustrates the exact one-to-one mapping between the given polynomial terms and the components of the cubic identity.

Step 4: Synthesizing the Factorised Form

Since all terms perfectly align with the expansion of $(x - y)^3$, we can rewrite the original polynomial in its expanded identity form:

$27 - 125a^3 - 135a + 225a^2 = (3)^3 - (5a)^3 - 3(3)^2(5a) + 3(3)(5a)^2$

Applying the identity $(x - y)^3$, where $x = 3$ and $y = 5a$, we condense the expression into a single perfect cube:

$= (3 - 5a)^3$

To express this as a product of its linear factors, we write the binomial three times.

Final Solution: The completely factorised form of the polynomial is $(3 - 5a)(3 - 5a)(3 - 5a)$.

Given Polynomial & Algebraic Foundation

We are tasked with factorising the following multivariable polynomial of degree 2:

$P(x,y,z) = 2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8xz$

The structure of this expression—comprising three squared terms and three cross-product terms—directly corresponds to the algebraic identity for the square of a trinomial. [Per the standard algebraic expansion theorem]:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Step 1: Extracting the Base Magnitudes

We begin by equating the pure squared terms from the given polynomial to the squared terms in the identity to find the absolute values of $a$, $b$, and $c$.

• $a^2 = 2x^2 \implies |a| = \sqrt{2}x$
• $b^2 = y^2 \implies |b| = y$
• $c^2 = 8z^2 \implies |c| = \sqrt{8}z = 2\sqrt{2}z$ [Since $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$]

Step 2: Determining the Algebraic Signs

To assign the correct positive ($+$) or negative ($-$) signs to $a$, $b$, and $c$, we must analyze the signs of the cross-product terms in the given polynomial:

From the analysis above, $b$ and $c$ share the same sign, while $a$ has the opposite sign to both. We can conventionally set $b$ and $c$ as positive, which forces $a$ to be negative. Therefore, we define our terms as:

• $a = -\sqrt{2}x$
• $b = y$
• $c = 2\sqrt{2}z$

Note: Alternatively, setting $a$ as positive forces $b$ and $c$ to be negative. Both conventions are mathematically equivalent.

Step 3: Verifying the Cross-Product Terms

Before finalizing the factorization, we must rigorously verify that our chosen terms reconstruct the original polynomial exactly.

• Check $2ab$: $2(-\sqrt{2}x)(y) = -2\sqrt{2}xy$ (Matches the given term)
• Check $2bc$: $2(y)(2\sqrt{2}z) = 4\sqrt{2}yz$ (Matches the given term)
• Check $2ca$: $2(2\sqrt{2}z)(-\sqrt{2}x) = -4(\sqrt{2} \cdot \sqrt{2})xz = -4(2)xz = -8xz$ (Matches the given term)

Step 4: Constructing the Factored Form

Since all terms perfectly align with the identity $(a + b + c)^2$, we substitute our derived values of $a$, $b$, and $c$ into the factored form.

$(a + b + c)^2 = (-\sqrt{2}x + y + 2\sqrt{2}z)^2$

To express this as a product of its factors, we write the squared binomial as the product of two identical trinomials.

Final Solution: The factorised form of the polynomial is $(-\sqrt{2}x + y + 2\sqrt{2}z)(-\sqrt{2}x + y + 2\sqrt{2}z)$.

(Note: Factoring out a $-1$ yields the equally valid alternative form: $(\sqrt{2}x - y - 2\sqrt{2}z)(\sqrt{2}x - y - 2\sqrt{2}z)$).

Given Expression & Initial Setup

We are tasked with factorising the following algebraic expression:

$64m^3 - 343n^3$

Upon initial inspection, the expression consists of two terms separated by a minus sign. The variables $m$ and $n$ are both raised to the third power, which strongly indicates that we must evaluate the numerical coefficients to determine if they are also perfect cubes.

Step 1: Identifying Perfect Cubes

To apply the relevant algebraic identity, we must rewrite both terms entirely as perfect cubes in the form of $x^3$ and $y^3$.

• First Term ($64m^3$): The prime factorization of $64$ is $2^6$, which can be grouped as $(2^2)^3 = 4^3$. Therefore, the entire first term can be rewritten as:
  $64m^3 = (4m)^3$
• Second Term ($343n^3$): The prime factorization of $343$ is $7 \times 7 \times 7 = 7^3$. Therefore, the entire second term can be rewritten as:
  $343n^3 = (7n)^3$

Substituting these back into the original expression yields:

$(4m)^3 - (7n)^3$

Step 2: Stating the Relevant Algebraic Identity

The expression is now explicitly in the form of a difference of two cubes. [Per the fundamental algebraic identities of polynomials], the difference of two cubes is factored using the following formula:

$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

Step 3: Substitution and Expansion

By mapping our specific terms to the identity, we establish the following equivalencies:

• $x = 4m$
• $y = 7n$

We now substitute these values directly into the right-hand side of the identity, $(x - y)(x^2 + xy + y^2)$:

$= (4m - 7n) \left[ (4m)^2 + (4m)(7n) + (7n)^2 \right]$

Step 4: Final Simplification

To achieve the final factorised form, we must expand the terms within the square brackets by applying the exponent rules [specifically $(ab)^n = a^n b^n$] and performing basic multiplication:

• Square the first term: $(4m)^2 = 4^2 \cdot m^2 = 16m^2$
• Multiply the two terms: $(4m)(7n) = 4 \cdot 7 \cdot m \cdot n = 28mn$
• Square the second term: $(7n)^2 = 7^2 \cdot n^2 = 49n^2$

Substituting these simplified components back into our factored expression yields:

$= (4m - 7n)(16m^2 + 28mn + 49n^2)$

Final Solution: $(4m - 7n)(16m^2 + 28mn + 49n^2)$

Initial Setup & Algebraic Analysis

We are tasked with factorising the following binomial expression:

$27y^3 + 125z^3$

To factorise this polynomial, we must first analyze the coefficients and the degrees of the variables to identify any underlying algebraic structures. We observe that both terms are perfect cubes.

Step 1: Expressing Terms as Perfect Cubes

We determine the cube roots of the numerical coefficients and the variables:

• For the first term: The prime factorization of $27$ is $3 \times 3 \times 3 = 3^3$. Therefore, $27y^3$ can be rewritten as $(3y)^3$ [Applying the power of a product rule: $x^n y^n = (xy)^n$].
• For the second term: The prime factorization of $125$ is $5 \times 5 \times 5 = 5^3$. Therefore, $125z^3$ can be rewritten as $(5z)^3$.

Rewriting the original expression, we get:

$(3y)^3 + (5z)^3$

Step 2: Stating the Relevant Algebraic Identity

The expression is now explicitly in the form of the sum of two cubes, $a^3 + b^3$. [Per the fundamental algebraic identity for the sum of cubes, derived from the expansion of $(a+b)^3 - 3ab(a+b)$], we know that:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Step 3: Substitution and Expansion

By mapping our terms to the identity, we set $a = 3y$ and $b = 5z$. Substituting these into the right-hand side of the identity yields:

$(3y)^3 + (5z)^3 = (3y + 5z) \left[ (3y)^2 - (3y)(5z) + (5z)^2 \right]$

Step 4: Simplification of the Quadratic Factor

We must now rigorously simplify each term inside the second set of parentheses (the quadratic trinomial factor):

• Square of the first term: $(3y)^2 = 3^2 \cdot y^2 = 9y^2$
• Product of the two terms: $(3y)(5z) = (3 \cdot 5)(y \cdot z) = 15yz$
• Square of the second term: $(5z)^2 = 5^2 \cdot z^2 = 25z^2$

Replacing the unsimplified terms in our expanded equation with these calculated values, we obtain the final factorised expression:

$(3y + 5z)(9y^2 - 15yz + 25z^2)$

Final Solution: The completely factorised form of $27y^3 + 125z^3$ is $(3y + 5z)(9y^2 - 15yz + 25z^2)$.

Step 1: Initial Setup and Polynomial Rearrangement

We are given the following polynomial expression of degree 3:

$P(a,b) = 64a^3 - 27b^3 - 144a^2b + 108ab^2$

To facilitate the identification of standard algebraic structures, we rearrange the terms in descending order of the powers of the first variable, $a$, and ascending order of the powers of the second variable, $b$.

$P(a,b) = 64a^3 - 144a^2b + 108ab^2 - 27b^3$

Step 2: Identification of the Relevant Algebraic Identity

The rearranged polynomial consists of four terms, with alternating signs ($+, -, +, -$), and the first and last terms appear to be perfect cubes. This structural signature strongly indicates the application of the standard binomial expansion identity for the cube of a difference [Per the Binomial Theorem for $(x-y)^n$ where $n=3$]:

$(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$

Step 3: Extracting the Base Terms ($x$ and $y$)

We equate the first and last terms of our polynomial to the corresponding terms in the identity to find the base variables $x$ and $y$.

• First Term ($x^3$):
  $x^3 = 64a^3$
  Taking the principal cube root of both sides:
  $x = \sqrt[3]{64a^3} = \sqrt[3]{4^3 \cdot a^3} = 4a$
• Last Term ($y^3$):
  $y^3 = 27b^3$
  Taking the principal cube root of both sides:
  $y = \sqrt[3]{27b^3} = \sqrt[3]{3^3 \cdot b^3} = 3b$

Step 4: Verification of the Cross Terms

To rigorously prove that the polynomial is indeed a perfect cube, we must verify that the middle terms of the given expression exactly match the $-3x^2y$ and $+3xy^2$ terms of the identity when $x = 4a$ and $y = 3b$.

• Verifying the second term ($-3x^2y$):
  $-3x^2y = -3(4a)^2(3b)$
  $= -3(16a^2)(3b)$
  $= -48a^2(3b) = -144a^2b$
  [This perfectly matches the second term of our rearranged polynomial.]
• Verifying the third term ($+3xy^2$):
  $+3xy^2 = 3(4a)(3b)^2$
  $= 3(4a)(9b^2)$
  $= 12a(9b^2) = 108ab^2$
  [This perfectly matches the third term of our rearranged polynomial.]

Step 5: Structural Mapping and Visualization

Since all four terms map perfectly to the $(x - y)^3$ identity, we can rewrite the entire polynomial in its unexpanded structural form.

$64a^3 - 144a^2b + 108ab^2 - 27b^3 = (4a)^3 - 3(4a)^2(3b) + 3(4a)(3b)^2 - (3b)^3$

Step 6: Final Factorization

By applying the identity $(x - y)^3$, we condense the expression into a single perfect cube:

$P(a,b) = (4a - 3b)^3$

To express this as a product of irreducible linear factors (which is the standard convention for complete factorization), we expand the exponent:

$P(a,b) = (4a - 3b)(4a - 3b)(4a - 3b)$

Final Solution: The completely factorised form of the polynomial $64a^3 - 27b^3 - 144a^2b + 108ab^2$ is $(4a - 3b)(4a - 3b)(4a - 3b)$.

Given Expression & Algebraic Identity

We are tasked with expanding the following trinomial squared:

$(3a - 7b - c)^2$

To expand this expression systematically, we utilize the standard algebraic identity for the square of a trinomial [Derived from the distributive property of multiplication over addition]:

$(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$

Step 1: Variable Mapping

We must map the terms of our specific expression to the variables in the standard identity. By rewriting the given expression as a sum, we ensure that the negative signs are correctly associated with their respective terms:

$(3a - 7b - c)^2 = (3a + (-7b) + (-c))^2$

By direct comparison with $(x + y + z)^2$, we establish the following variable assignments:

• $x = 3a$
• $y = -7b$
• $z = -c$

Step 2: Substitution into the Identity

Substituting these mapped values into the expanded form of the identity yields:

$(3a + (-7b) + (-c))^2 = (3a)^2 + (-7b)^2 + (-c)^2 + 2(3a)(-7b) + 2(-7b)(-c) + 2(-c)(3a)$

Step 3: Term-by-Term Simplification

We now evaluate each term individually, applying the rules of exponents [$(ab)^n = a^n b^n$] and the rules of sign multiplication:

• Squaring the first term: $(3a)^2 = 3^2 \cdot a^2 = 9a^2$
• Squaring the second term: $(-7b)^2 = (-7)^2 \cdot b^2 = 49b^2$ [Note: The square of a negative number is positive]
• Squaring the third term: $(-c)^2 = (-1)^2 \cdot c^2 = c^2$
• First cross-product: $2(3a)(-7b) = 2 \cdot 3 \cdot (-7) \cdot a \cdot b = -42ab$
• Second cross-product: $2(-7b)(-c) = 2 \cdot (-7) \cdot (-1) \cdot b \cdot c = 14bc$ [Note: The product of two negative numbers is positive]
• Third cross-product: $2(-c)(3a) = 2 \cdot (-1) \cdot 3 \cdot c \cdot a = -6ca$

Step 4: Assembling the Final Polynomial

Combining all the simplified terms from Step 3, we construct the final expanded polynomial:

$9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ca$

Geometric Representation of the Trinomial Square Identity

The algebraic identity $(x+y+z)^2$ can be visualized as the area of a square with side length $(x+y+z)$, partitioned into 9 distinct rectangular regions. The sum of the areas of these regions corresponds exactly to the terms in our expanded formula.

Final Solution: $9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ca$

Given Variables & Theoretical Foundation

We are given the volume of a cuboid expressed as a polynomial in terms of a variable $x$:

$V(x) = 3x^2 - 12x$

[Per the geometric definition of a cuboid], the volume $V$ is the product of its three mutually perpendicular spatial dimensions: Length ($L$), Width ($W$), and Height ($H$). Mathematically, this is expressed as:

$V = L \times W \times H$

To find the possible expressions for the dimensions of the cuboid, we must factorize the given binomial polynomial into three distinct linear or constant factors.

Step 1: Identifying the Greatest Common Factor (GCF)

We begin by analyzing the terms of the polynomial $3x^2 - 12x$ to extract the Greatest Common Factor (GCF). We decompose each term into its prime numerical and algebraic factors.

By comparing the factorizations, we identify the common elements:

• Numerical GCF: The greatest integer that divides both $3$ and $-12$ is $3$.
• Algebraic GCF: The highest power of $x$ common to both $x^2$ and $x$ is $x^1$ (or simply $x$).

Therefore, the overall Greatest Common Factor is $3x$.

Step 2: Algebraic Factorization

[By the Distributive Property of Multiplication over Addition], we can factor out the GCF from the original expression:

$V(x) = 3x^2 - 12x$

$V(x) = 3x(x) - 3x(4)$

$V(x) = 3x(x - 4)$

We have now successfully expressed the binomial as a product of its irreducible factors.

Step 3: Mapping Factors to Geometric Dimensions

The factored form of the volume is $3 \cdot x \cdot (x - 4)$. Because the volume of a cuboid requires three dimensions ($L \times W \times H$), we can directly map these three distinct factors to the dimensions of the cuboid.

• Dimension 1: $3$
• Dimension 2: $x$
• Dimension 3: $(x - 4)$

Note: Because multiplication is commutative ($A \times B \times C = B \times C \times A$), any of these expressions can represent the length, width, or height.

Geometric Visualization

Below is a high-precision spatial representation of the cuboid with its corresponding dimensional expressions.

Final Solution: The possible expressions for the dimensions of the cuboid are $3$, $x$, and $(x - 4)$.

Given Expression & Algebraic Identity

We are tasked with expanding the following trinomial squared:

$(–2x + 3y + 2z)^2$

To expand this expression systematically, we utilize the standard algebraic identity for the square of a trinomial [Derived from the distributive property of multiplication over addition]:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Step 1: Variable Mapping

By comparing the given expression $(–2x + 3y + 2z)^2$ with the standard identity $(a + b + c)^2$, we establish the following one-to-one correspondence for the terms:

• $a = -2x$
• $b = 3y$
• $c = 2z$

Note: It is critical to include the negative sign in the assignment of $a$ to ensure the cross-terms are calculated with the correct parity.

Step 2: Substitution into the Identity

Substituting the mapped variables into the expanded form of the identity yields:

$(-2x + 3y + 2z)^2 = (-2x)^2 + (3y)^2 + (2z)^2 + 2(-2x)(3y) + 2(3y)(2z) + 2(2z)(-2x)$

Step 3: Term-by-Term Expansion & Simplification

We now evaluate each term individually, applying the rules of exponents [$(xy)^n = x^n y^n$] and the rules of signed multiplication:

Step 4: Final Aggregation

Combining all the simplified terms from Step 3 into a single polynomial expression:

$4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8zx$

Geometric Visualization of the Identity

The algebraic identity $(a+b+c)^2$ can be geometrically proven by calculating the area of a square with side length $(a+b+c)$. The total area is the sum of the 9 smaller rectangular regions, which perfectly mirrors our algebraic expansion.

Final Solution: The expanded form of $(-2x + 3y + 2z)^2$ is $4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8zx$.

Initial Setup & Algebraic Objective

We are tasked with verifying the following fundamental algebraic identity:

$x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$

To prove this rigorously, we will operate on the Right-Hand Side (RHS) of the equation and demonstrate, through sequential expansion and simplification, that it is mathematically equivalent to the Left-Hand Side (LHS).

Step 1: Expansion of the Squared Binomials on the RHS

Consider the Right-Hand Side (RHS) of the given equation:

$\text{RHS} = \frac{1}{2}(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]$

We begin by expanding the three squared binomial terms inside the square brackets. [Per the standard algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$], we obtain:

• $(x - y)^2 = x^2 - 2xy + y^2$
• $(y - z)^2 = y^2 - 2yz + z^2$
• $(z - x)^2 = z^2 - 2zx + x^2$

Step 2: Aggregation and Simplification of the Expanded Terms

Substitute these expanded forms back into the bracketed expression:

$[(x^2 - 2xy + y^2) + (y^2 - 2yz + z^2) + (z^2 - 2zx + x^2)]$

Next, we group the like terms (the squared variables):

$= (x^2 + x^2) + (y^2 + y^2) + (z^2 + z^2) - 2xy - 2yz - 2zx$

$= 2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2zx$

By factoring out the greatest common scalar multiplier ($2$), the expression simplifies to:

$= 2(x^2 + y^2 + z^2 - xy - yz - zx)$

Step 3: Cancellation of the Scalar Multiplier

Now, substitute this simplified bracketed expression back into the full RHS equation:

$\text{RHS} = \frac{1}{2}(x + y + z) \cdot \left[ 2(x^2 + y^2 + z^2 - xy - yz - zx) \right]$

The scalar fraction $\frac{1}{2}$ and the factored integer $2$ cancel each other out exactly ($\frac{1}{2} \times 2 = 1$), yielding:

$\text{RHS} = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$

Note: At this stage, the expression matches the standard factorization of $x^3 + y^3 + z^3 - 3xyz$. However, for absolute rigor, we will perform the full polynomial distribution to prove the equivalence to the LHS.

Step 4: Final Polynomial Distribution

We distribute the trinomial $(x + y + z)$ across the polynomial $(x^2 + y^2 + z^2 - xy - yz - zx)$ by multiplying each term in the first bracket by every term in the second bracket:

Distributing $x$:
$x(x^2 + y^2 + z^2 - xy - yz - zx) = x^3 + xy^2 + xz^2 - x^2y - xyz - zx^2$

Distributing $y$:
$y(x^2 + y^2 + z^2 - xy - yz - zx) = x^2y + y^3 + yz^2 - xy^2 - y^2z - xyz$

Distributing $z$:
$z(x^2 + y^2 + z^2 - xy - yz - zx) = zx^2 + zy^2 + z^3 - xyz - yz^2 - xz^2$

Now, we sum all the distributed terms together:

$= x^3 + y^3 + z^3$
$\quad + xy^2 - xy^2$
$\quad + xz^2 - xz^2$
$\quad - x^2y + x^2y$
$\quad - zx^2 + zx^2$
$\quad + yz^2 - yz^2$
$\quad - y^2z + zy^2$
$\quad - xyz - xyz - xyz$

Observe the systematic cancellation of the intermediate cross-terms:

• $xy^2$ cancels with $-xy^2$
• $xz^2$ cancels with $-xz^2$
• $-x^2y$ cancels with $x^2y$
• $-zx^2$ cancels with $zx^2$
• $yz^2$ cancels with $-yz^2$
• $-y^2z$ cancels with $zy^2$ (since $zy^2 = y^2z$)

The only terms that survive the cancellation are the cubic terms and the three $-xyz$ terms:

$= x^3 + y^3 + z^3 - 3xyz$

This resulting expression is exactly the Left-Hand Side (LHS) of our initial equation.

Final Solution: By expanding the right-hand side and systematically canceling the intermediate polynomial terms, we have rigorously proven that $\text{RHS} = \text{LHS}$. Therefore, the identity $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$ is verified.

Step 1: Algebraic Formulation and Base Selection

To evaluate the product $104 \times 96$ without performing direct arithmetic multiplication, we must express both factors in terms of a common, easily computable base. Observing the numerical values, both $104$ and $96$ are symmetrically distributed around the base value of $100$.

• The first factor can be written as an addition to the base: $104 = 100 + 4$
• The second factor can be written as a subtraction from the base: $96 = 100 - 4$

Substituting these expressions back into the original product, we obtain:

$104 \times 96 = (100 + 4)(100 - 4)$

Step 2: Application of the Difference of Squares Identity

The formulated expression $(100 + 4)(100 - 4)$ perfectly matches the structure of a fundamental polynomial identity [Per the algebraic identity for the product of the sum and difference of two terms].

The standard identity is defined as:

$(a + b)(a - b) = a^2 - b^2$

By mapping our specific variables to the identity, we establish:

• $a = 100$
• $b = 4$

Applying the identity to our expression yields:

$(100 + 4)(100 - 4) = (100)^2 - (4)^2$

Step 3: Geometric Visualization of the Identity

The algebraic identity $(a+b)(a-b) = a^2 - b^2$ can be rigorously proven through geometric area conservation. The area of a rectangle with dimensions $(a+b)$ and $(a-b)$ is mathematically equivalent to the area of a large square of side $a$ minus the area of a smaller square of side $b$.

Step 4: Computation of Squares and Final Evaluation

We now compute the numerical values of the squared terms derived in Step 2:

• Square of the base term: $(100)^2 = 100 \times 100 = 10000$
• Square of the deviation term: $(4)^2 = 4 \times 4 = 16$

Substitute these calculated values back into the difference equation:

$10000 - 16$

Performing the final subtraction operation:

$10000 - 16 = 9984$

Final Solution: 9984

Initial Setup & Theoretical Foundation

The geometric area of a rectangle is defined by the product of its two adjacent spatial dimensions, length and breadth. Mathematically, this is expressed as:

[$Area = Length \times Breadth$]

We are given the area of a rectangle as a quadratic polynomial in terms of the variable $y$:

$Area = 35y^2 + 13y - 12$

To find the possible expressions for the length and breadth, we must factorize this quadratic polynomial into the product of two linear binomials. Each resulting binomial will represent one of the dimensions of the rectangle.

Step 1: Coefficient Analysis and the Product-Sum Method

We will factorize the quadratic polynomial $ay^2 + by + c$ using the method of splitting the middle term. First, we identify the coefficients:

• Leading coefficient ($a$) = $35$
• Middle coefficient ($b$) = $13$
• Constant term ($c$) = $-12$

According to the product-sum factorization theorem, we must find two real numbers, let's call them $p$ and $q$, such that:

1. Their product equals $a \times c$:
$p \times q = 35 \times (-12) = -420$

2. Their sum equals $b$:
$p + q = 13$

Step 2: Prime Factorization to Determine Split Values

To systematically find the values of $p$ and $q$, we analyze the prime factorization of the absolute value of the product ($420$):

$420 = 2 \times 210 = 2^2 \times 105 = 2^2 \times 3 \times 35 = 2^2 \times 3 \times 5 \times 7$

We need to group these prime factors into two numbers whose difference is $13$ (since the product is negative, one number must be positive and the other negative). Let us test combinations:

• Combination 1: $(2^2 \times 5) = 20$ and $(3 \times 7) = 21$. Difference is $1$. (Incorrect)
• Combination 2: $(2^2 \times 7) = 28$ and $(3 \times 5) = 15$. Difference is $13$. (Correct)

Since the sum must be positive ($+13$), the larger number must be positive. Therefore, our two numbers are $28$ and $-15$.

[$28 \times (-15) = -420$ and $28 + (-15) = 13$]

Step 3: Splitting the Middle Term and Grouping

We substitute the middle term $13y$ with $28y - 15y$ in the original polynomial:

$35y^2 + 28y - 15y - 12$

Next, we apply factorization by grouping. We group the first two terms and the last two terms:

$(35y^2 + 28y) - (15y + 12)$

Extract the greatest common divisor (GCD) from each group:

For $(35y^2 + 28y)$, the GCD is $7y$:
$7y(5y + 4)$

For $-(15y + 12)$, the GCD is $-3$:
$-3(5y + 4)$

Now, substitute these back into the expression:

$7y(5y + 4) - 3(5y + 4)$

Notice that the binomial $(5y + 4)$ is a common factor. Factoring it out yields the final product of two linear expressions:

$(7y - 3)(5y + 4)$

Step 4: Geometric Representation and Dimensional Assignment

Because multiplication is commutative [by the Commutative Property of Multiplication, $A \times B = B \times A$], either of these binomial factors can represent the length, and the other will represent the breadth.

Final Solution: The possible expressions for the dimensions of the rectangle are Length = $(7y - 3)$ and Breadth = $(5y + 4)$, or vice versa.

Given Variables & Initial Setup

We are given the linear equation involving three variables:

$x + y + z = 0$

Objective: Prove the algebraic relationship $x^3 + y^3 + z^3 = 3xyz$.

To provide a comprehensive, masterclass-level proof, this relationship will be demonstrated using two distinct mathematical approaches: the Standard Identity Method and the Direct Algebraic Manipulation (Cubing) Method.

Method 1: Utilizing the Standard Algebraic Identity

Step 1: State the relevant polynomial identity

In algebra, the sum of three cubes is governed by the following fundamental identity:

$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$

[Theoretical Justification: This identity is derived by expanding the right-hand side and canceling out the intermediate cross-terms, leaving only the sum of the cubes and the $-3xyz$ term.]

Step 2: Substitute the given condition

We are given the premise that $x + y + z = 0$. We substitute this value directly into the right-hand side of our identity:

$x^3 + y^3 + z^3 - 3xyz = (0) \cdot (x^2 + y^2 + z^2 - xy - yz - zx)$

Step 3: Apply the Zero Product Property

According to the Zero Product Property, any finite real number or algebraic expression multiplied by zero results in zero. Therefore, the entire right-hand side collapses to $0$:

$x^3 + y^3 + z^3 - 3xyz = 0$

Step 4: Isolate the sum of the cubes

By adding $3xyz$ to both sides of the equation, we arrive at the final required expression:

$x^3 + y^3 + z^3 = 3xyz$

Method 2: Direct Algebraic Manipulation (Cubing)

Step 1: Rearrange the initial equation

Starting with the given condition, isolate two variables on one side of the equation:

$x + y + z = 0 \implies x + y = -z$

Step 2: Cube both sides of the equation

To generate the cubic terms required for the proof, apply the power of 3 to both sides:

$(x + y)^3 = (-z)^3$

Step 3: Expand using the binomial cube identity

Expand the left side using the standard binomial identity $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$. Note that the cube of a negative value remains negative, so $(-z)^3 = -z^3$.

$x^3 + y^3 + 3xy(x + y) = -z^3$

Step 4: Substitute the initial rearranged condition

From Step 1, we established that $(x + y) = -z$. Substitute $-z$ back into the expanded equation in place of $(x + y)$:

$x^3 + y^3 + 3xy(-z) = -z^3$

Step 5: Simplify and rearrange terms

Multiply the terms to simplify the equation:

$x^3 + y^3 - 3xyz = -z^3$

Finally, transpose $-z^3$ to the left side (becoming $+z^3$) and $-3xyz$ to the right side (becoming $+3xyz$):

$x^3 + y^3 + z^3 = 3xyz$

Visual Representation of the Logical Pathways

The following flowchart illustrates the dual algebraic pathways utilized to prove the theorem, confirming the structural integrity of both methods.

Final Conclusion

Both the application of the standard cubic polynomial identity and direct algebraic expansion yield the exact same mathematical truth. When the sum of three variables is zero, the sum of their cubes is perfectly balanced by three times their product.

Final Solution: It is proven that if $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$.

Given Expression & Algebraic Identity

We are tasked with expanding the following algebraic expression:

$(2x - y + z)^2$

To expand a trinomial squared, we utilize the standard algebraic identity for the square of a trinomial [derived via the distributive property of multiplication over addition]:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Step 1: Variable Mapping

By comparing the given expression $(2x - y + z)^2$ with the standard identity $(a + b + c)^2$, we establish a direct mapping of the terms. It is critical to include the negative sign in the mapping to maintain algebraic equivalence.

• $a = 2x$
• $b = -y$
• $c = z$

Step 2: Term-by-Term Expansion

Substitute the mapped variables into the expanded form of the identity. We will compute the squared terms and the cross-product terms systematically.

1. Computing the Squared Terms ($a^2, b^2, c^2$):

• $a^2 = (2x)^2 = 4x^2$ [Applying the power of a product rule: $(xy)^n = x^n y^n$]
• $b^2 = (-y)^2 = y^2$ [The square of a negative quantity is positive]
• $c^2 = (z)^2 = z^2$

2. Computing the Cross-Product Terms ($2ab, 2bc, 2ca$):

• $2ab = 2(2x)(-y) = -4xy$
• $2bc = 2(-y)(z) = -2yz$
• $2ca = 2(z)(2x) = 4zx$

Step 3: Synthesis and Simplification

Now, we assemble all the computed components back into the framework of the identity:

$(2x - y + z)^2 = (4x^2) + (y^2) + (z^2) + (-4xy) + (-2yz) + (4zx)$

Simplifying the signs yields the final expanded polynomial:

$(2x - y + z)^2 = 4x^2 + y^2 + z^2 - 4xy - 2yz + 4zx$

Geometric Interpretation of the Identity

The algebraic identity $(a+b+c)^2$ can be visualized geometrically as the area of a square with side length $(a+b+c)$. The total area is the sum of the areas of the 9 smaller rectangular regions formed by partitioning the sides into segments of lengths $a$, $b$, and $c$.

As demonstrated in the area model, the total area consists of one $a^2$, one $b^2$, one $c^2$, two $ab$ rectangles, two $bc$ rectangles, and two $ac$ rectangles, perfectly mirroring our algebraic expansion.

Final Solution: The expanded form of $(2x - y + z)^2$ is $4x^2 + y^2 + z^2 - 4xy - 2yz + 4zx$.

Initial Setup & Expression Analysis

We are tasked with factorising the following algebraic polynomial:

$9x^2 + 6xy + y^2$

To factorise this expression efficiently, we must analyze its structural properties. The polynomial consists of three terms (a trinomial). Our primary objective is to determine if it conforms to the structure of a perfect square trinomial, which can be factorised using standard algebraic identities.

Step 1: Decomposing the Terms into Perfect Squares

We begin by examining the first and third terms to see if they can be expressed as perfect squares [Per the fundamental properties of exponents].

• First Term: The term $9x^2$ consists of a constant $9$ and a variable $x^2$. Since $3^2 = 9$, we can rewrite this term as the square of a single monomial:
  $9x^2 = (3x)^2$
• Third Term: The term $y^2$ is already a perfect square and can be explicitly written as:
  $y^2 = (y)^2$

Step 2: Verifying the Middle Term

For a trinomial to be classified as a perfect square trinomial, the middle term must be exactly twice the product of the bases of the squared terms identified in Step 1.

Let us define our base variables based on the standard identity structure:

• Let $a = 3x$
• Let $b = y$

Now, we calculate the theoretical middle term, $2ab$:

$2ab = 2 \cdot (3x) \cdot (y)$

$2ab = 6xy$

This calculated product perfectly matches the middle term of our given polynomial ($6xy$). Therefore, the expression is confirmed to be a perfect square trinomial.

Step 3: Applying the Standard Algebraic Identity

Having verified the structure, we apply the First Algebraic Identity for the square of a binomial [By the binomial expansion theorem]:

$a^2 + 2ab + b^2 = (a + b)^2$

Substituting our specific values ($a = 3x$ and $b = y$) into the identity:

$(3x)^2 + 2(3x)(y) + (y)^2 = (3x + y)^2$

Expanding the squared binomial into its linear factors yields:

$(3x + y)^2 = (3x + y)(3x + y)$

Geometric Proof & Visualization

The algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$ can be rigorously proven using area models in Euclidean geometry. Below is the geometric representation of our specific polynomial, where the total area of a square with side length $(3x + y)$ is equal to the sum of the areas of its four internal rectangular regions.

As demonstrated by the geometric model, the total area is the sum of the individual regions: $9x^2 + 3xy + 3xy + y^2 = 9x^2 + 6xy + y^2$, which perfectly corresponds to a square of side $(3x + y)$.

Final Solution: The completely factorised form of the polynomial $9x^2 + 6xy + y^2$ is $(3x + y)(3x + y)$, which is most concisely written as $(3x + y)^2$.

Step 1: Given Variables & Initial Setup

We are tasked with evaluating the sum of three cubes without performing direct cubing operations. The given mathematical expression is:

$(-12)^3 + (7)^3 + (5)^3$

Let us define the base of each cubic term as a distinct variable:

• $a = -12$
• $b = 7$
• $c = 5$

Step 2: Evaluating the Sum of the Base Variables

Before attempting to expand the expression, we must analyze the linear sum of the base variables. This is a critical diagnostic step in polynomial algebra to determine if a conditional identity applies.

$a + b + c = (-12) + 7 + 5$

$a + b + c = -12 + 12 = 0$

[Geometrically, this represents a closed vector loop in one dimension, where the displacement from the origin returns exactly to zero. See the precise vector visualization below.]

Step 3: Theoretical Justification & Algebraic Identity

We rely on the fundamental algebraic identity for the sum of three cubes:

$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

[Per the Zero Product Property], if the first factor on the right-hand side, $(a + b + c)$, is equal to zero, the entire right-hand side of the equation evaluates to zero, regardless of the values of the quadratic polynomial factor. Therefore:

$a^3 + b^3 + c^3 - 3abc = (0) \times (a^2 + b^2 + c^2 - ab - bc - ca)$

$a^3 + b^3 + c^3 - 3abc = 0$

Transposing $-3abc$ to the right side yields the conditional identity:

If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.

Step 4: Application of the Conditional Identity

Since we have rigorously proven in Step 2 that $(-12) + 7 + 5 = 0$, we can directly substitute our variables into the conditional identity:

$(-12)^3 + (7)^3 + (5)^3 = 3 \cdot (-12) \cdot (7) \cdot (5)$

Step 5: Final Arithmetic Calculation

We now perform the sequential multiplication of the terms on the right-hand side:

• First, multiply the constants: $3 \times (-12) = -36$
• Next, multiply the remaining terms: $7 \times 5 = 35$
• Finally, compute the product of the two results: $(-36) \times 35$

To calculate $(-36) \times 35$ mentally or systematically:

$(-36) \times (30 + 5) = (-36 \times 30) + (-36 \times 5)$

$= -1080 - 180 = -1260$

Final Solution: The value of $(-12)^3 + (7)^3 + (5)^3$ is $-1260$.

Initial Setup & Algebraic Identity

We are tasked with expanding the algebraic expression $(x + 2y + 4z)^2$. To execute this expansion systematically, we utilize the standard algebraic identity for the square of a trinomial.

[Per the distributive property of multiplication over addition, the square of a trinomial is given by the identity:]

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Step 1: Variable Mapping

By comparing our given expression $(x + 2y + 4z)^2$ with the standard identity $(a + b + c)^2$, we establish a direct one-to-one mapping of the terms:

• $a = x$
• $b = 2y$
• $c = 4z$

Geometric Visualization of the Identity

The algebraic expansion of $(a + b + c)^2$ can be geometrically interpreted as the area of a square with side length $(a + b + c)$, partitioned into nine distinct rectangular and square regions. The sum of the areas of these nine regions perfectly mirrors the terms in our algebraic identity.

Step 2: Substitution into the Identity

Substituting the mapped variables into the right-hand side of the identity, we obtain the unsimplified expanded form:

$(x + 2y + 4z)^2 = (x)^2 + (2y)^2 + (4z)^2 + 2(x)(2y) + 2(2y)(4z) + 2(4z)(x)$

Step 3: Term-by-Term Simplification

We now apply the laws of exponents [specifically $(xy)^n = x^n y^n$] and basic arithmetic multiplication to simplify each term independently.

Step 4: Final Assembly

Combining all the simplified terms from Step 3 yields the fully expanded polynomial. By convention, we write the squared terms first, followed by the cross-product terms in cyclical order ($xy$, $yz$, $zx$).

$(x + 2y + 4z)^2 = x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8zx$

Final Solution: The expanded form of $(x + 2y + 4z)^2$ is $x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8zx$.