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 $-1260$.

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

Initial Setup & Given Expression

We are tasked with factorising the following algebraic expression:

$ 27p^3 - \frac{1}{216} - \frac{9}{2}p^2 + \frac{1}{4}p $

To systematically approach the factorisation, we first rearrange the terms in descending order of the powers of the variable $p$ [Per standard polynomial representation conventions]:

$ 27p^3 - \frac{9}{2}p^2 + \frac{1}{4}p - \frac{1}{216} $

Step 1: Identifying the Applicable Algebraic Identity

The expression consists of four terms, beginning and ending with perfect cubes. This structural signature strongly indicates the expansion of the binomial cube identity. We recall the standard algebraic identity for the cube of a difference:

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

Step 2: Extracting the Base Variables ($a$ and $b$)

We analyze the first and last terms of our rearranged polynomial to determine the values of $a$ and $b$.

• First Term ($a^3$): The term $27p^3$ can be written as a perfect cube. Since $3^3 = 27$, we have:
  $ a^3 = (3p)^3 \implies a = 3p $
• Last Term ($b^3$): The term $\frac{1}{216}$ can be written as a perfect cube. Since $6^3 = 216$, we have:
  $ b^3 = \left(\frac{1}{6}\right)^3 \implies b = \frac{1}{6} $

Step 3: Verifying the Middle Terms

To rigorously prove that the expression is indeed the expansion of $(3p - \frac{1}{6})^3$, we must substitute $a = 3p$ and $b = \frac{1}{6}$ into the middle terms of the identity ($-3a^2b$ and $+3ab^2$) and verify that they match the original polynomial.

Checking the second term ($-3a^2b$):

$ -3a^2b = -3(3p)^2\left(\frac{1}{6}\right) $

$ = -3(9p^2)\left(\frac{1}{6}\right) $

$ = -27p^2 \left(\frac{1}{6}\right) $

$ = -\frac{27}{6}p^2 $

Simplifying the fraction by dividing the numerator and denominator by 3:

$ = -\frac{9}{2}p^2 $

[This perfectly matches the second term of our rearranged polynomial.]

Checking the third term ($+3ab^2$):

$ +3ab^2 = 3(3p)\left(\frac{1}{6}\right)^2 $

$ = 9p\left(\frac{1}{36}\right) $

$ = \frac{9}{36}p $

Simplifying the fraction by dividing the numerator and denominator by 9:

$ = \frac{1}{4}p $

[This perfectly matches the third term of our rearranged polynomial.]

Step 4: Constructing the Factorised Form

Since all four terms of the given polynomial perfectly map to the expansion of $(a - b)^3$, we can confidently compress the expression into its factorised binomial cube form.

Substituting $a = 3p$ and $b = \frac{1}{6}$ into $(a - b)^3$ yields:

$ \left(3p - \frac{1}{6}\right)^3 $

To express this as a complete factorisation (a product of irreducible polynomials), we write the binomial three times:

$ \left(3p - \frac{1}{6}\right) \left(3p - \frac{1}{6}\right) \left(3p - \frac{1}{6}\right) $

Final Solution: The fully factorised form of the given polynomial is $ \left(3p - \frac{1}{6}\right) \left(3p - \frac{1}{6}\right) \left(3p - \frac{1}{6}\right) $.

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

Given Variables & Initial Setup

The volume of a cuboid is defined by the product of its three mutually perpendicular dimensions: length ($l$), width ($w$), and height ($h$).

Mathematically, this is expressed as:
$V = l \times w \times h$    [Per the geometric definition of a rectangular prism's volume]

We are given the volume of a cuboid as a polynomial expression in terms of variables $k$ and $y$:

$V(y) = 12ky^2 + 8ky - 20k$

To find the possible expressions for the dimensions, we must factorize this polynomial into three distinct linear factors, which will correspond to the length, width, and height.

Step 1: Extracting the Greatest Common Factor (GCF)

We begin by analyzing the terms of the polynomial: $12ky^2$, $8ky$, and $-20k$. We look for the highest common numerical coefficient and the highest degree of common variables.

• The numerical coefficients are $12$, $8$, and $-20$. The greatest common divisor (GCD) of these integers is $4$.
• Each term contains the variable $k$ to the first power.
• The variable $y$ is not present in the third term, so it cannot be factored out.

Factoring out the GCF ($4k$) from the entire expression yields:

$V(y) = 4k \left( \frac{12ky^2}{4k} + \frac{8ky}{4k} - \frac{20k}{4k} \right)$

$V(y) = 4k(3y^2 + 2y - 5)$

Step 2: Factorizing the Quadratic Expression

The expression inside the parentheses is a quadratic polynomial of the form $ay^2 + by + c$, where $a = 3$, $b = 2$, and $c = -5$. We will factorize this using the method of splitting the middle term.

We must find two numbers that satisfy two conditions simultaneously:

1. Their product equals $a \times c = 3 \times (-5) = -15$.
2. Their sum equals the middle coefficient $b = 2$.

The factors of $-15$ that add up to $2$ are $5$ and $-3$. We rewrite the middle term ($2y$) using these two numbers:

$3y^2 + 5y - 3y - 5$

Next, we group the terms into pairs to factor by grouping:

$(3y^2 - 3y) + (5y - 5)$

Factor out the common terms from each binomial group:

$3y(y - 1) + 5(y - 1)$

Since $(y - 1)$ is a common binomial factor, we can factor it out:

$(3y + 5)(y - 1)$

Step 3: Determining the Dimensions

Substituting the factorized quadratic back into our volume equation, we get the completely factorized form of the volume:

$V = 4k \cdot (3y + 5) \cdot (y - 1)$

Because the volume of a cuboid is the product of three dimensions ($l \times w \times h$), these three factors represent the possible expressions for the length, width, and height of the cuboid.

Geometric Visualization

Below is a precise oblique projection of the cuboid, illustrating how the three algebraic factors map to the spatial dimensions.

Final Solution: The possible expressions for the dimensions of the cuboid are $4k$, $(3y + 5)$, and $(y - 1)$.

Initial Setup & Algebraic Formulation

We are tasked with evaluating the numerical expression $(99)^3$ without relying on direct, brute-force multiplication. To achieve this with mathematical rigor, we must utilize standard algebraic identities.

Step 1: Binomial Transformation

To apply an algebraic identity, we must first express the base number, $99$, as a binomial. The most computationally efficient approach is to express $99$ as the difference between a power of $10$ and an integer, as powers of $10$ are trivial to exponentiate.

$99 = 100 - 1$

Therefore, the expression becomes:

$(99)^3 = (100 - 1)^3$

Step 2: Selection of the Algebraic Identity

The expression $(100 - 1)^3$ perfectly matches the standard binomial expansion identity for the cube of a difference [Per the Binomial Theorem for $(a-b)^n$ where $n=3$]. The identity is defined as:

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

Note: This is the expanded form of $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$, which allows for clear, term-by-term evaluation.

Step 3: Substitution of Variables

By mapping our binomial to the identity, we establish the following variable assignments:

• $a = 100$
• $b = 1$

Substituting these specific values into the expanded identity yields:

$(100 - 1)^3 = (100)^3 - 3(100)^2(1) + 3(100)(1)^2 - (1)^3$

Step 4: Term-by-Term Evaluation

We now compute each of the four terms individually to ensure absolute arithmetic precision.

Visualizing the Binomial Expansion

The following flowchart maps the distribution of the binomial cube into its constituent polynomial terms before final synthesis.

Step 5: Final Arithmetic Synthesis

Substituting the evaluated terms back into the expanded equation, we perform the final addition and subtraction sequentially from left to right:

$(99)^3 = 1,000,000 - 30,000 + 300 - 1$

First, subtract $30,000$ from $1,000,000$:

$1,000,000 - 30,000 = 970,000$

Next, add $300$:

$970,000 + 300 = 970,300$

Finally, subtract $1$:

$970,300 - 1 = 970,299$

Final Solution: $(99)^3 = 970,299$

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

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 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 $(4a - 3b)(4a - 3b)(4a - 3b)$.

Initial Algebraic Setup

We are tasked with factorising the following algebraic expression:

$8a^3 + b^3 + 12a^2b + 6ab^2$

Step 1: Identifying the Relevant Algebraic Identity

By analyzing the structure of the polynomial, we observe that it contains four terms, all of which are positive. Furthermore, the first and second terms ($8a^3$ and $b^3$) are perfect cubes. This structural signature strongly indicates the application of the standard binomial expansion identity for the cube of a sum [Per the Binomial Theorem for exponent 3].

The standard identity is defined as:

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

Our objective is to map the given polynomial to the right-hand side of this identity.

Step 2: Decomposing the Terms to Establish Base Variables

We must rewrite the perfect cube terms to identify the base variables $x$ and $y$.

• First Term (Cube): The term $8a^3$ can be rewritten as the cube of a single monomial. Since $2^3 = 8$, we have:
  $8a^3 = (2a)^3$
  [This establishes our $x = 2a$].
• Second Term (Cube): The term $b^3$ is already a perfect cube:
  $b^3 = (b)^3$
  [This establishes our $y = b$].

Step 3: Verifying the Cross-Terms

To rigorously prove that the identity applies, we must verify that the remaining terms in the polynomial ($12a^2b$ and $6ab^2$) perfectly match the $3x^2y$ and $3xy^2$ components of the identity using our established $x = 2a$ and $y = b$.

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

Step 4: Visualizing the Algebraic Mapping

The following diagram illustrates the exact one-to-one mapping between the standard identity and our decomposed polynomial.

Step 5: Applying the Identity to Factorise

Since all terms perfectly satisfy the expansion of $(x + y)^3$, we can condense the expanded polynomial into its factorised binomial cube form by substituting $x = 2a$ and $y = b$:

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

Step 6: Expanding into Linear Factors

To express the polynomial fully factorised into irreducible linear polynomials, we expand the exponent. [By the fundamental definition of exponents, $z^3 = z \cdot z \cdot z$].

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

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

Initial Setup & Theoretical Foundation

The area of a rectangle is geometrically defined as the product of its two adjacent spatial dimensions: length ($l$) and breadth ($b$). This relationship is expressed by the formula:

$\text{Area} = l \times b$

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

$\text{Area} = 25a^2 - 35a + 12$

[Per the Fundamental Theorem of Algebra and polynomial factorization principles], to find the possible expressions for the length and breadth, we must factorize this quadratic polynomial into the product of two linear binomials. These two resulting factors will represent the possible dimensions of the rectangle.

Step 1: Identifying the Factorization Method

We will factorize the quadratic polynomial $P(a) = 25a^2 - 35a + 12$ using the method of splitting the middle term. We compare the given polynomial to the standard quadratic form $Ax^2 + Bx + C$ to identify the coefficients:

• $A = 25$ (Coefficient of $a^2$)
• $B = -35$ (Coefficient of $a$)
• $C = 12$ (Constant term)

Step 2: Splitting the Middle Term

To split the middle term, we must find two real numbers, let us call them $p$ and $q$, that satisfy two specific conditions simultaneously:

1. Their product must equal $A \times C$:
   $p \times q = 25 \times 12 = 300$
2. Their sum must equal $B$:
   $p + q = -35$

Because the product ($300$) is positive and the sum ($-35$) is negative, [by the rules of integer arithmetic], both numbers $p$ and $q$ must be negative.

Let us evaluate the factor pairs of $300$ to find the correct combination:

The numbers $-15$ and $-20$ satisfy both conditions perfectly. Therefore, we will split the middle term $-35a$ into $-20a$ and $-15a$.

Step 3: Factorization by Grouping

We substitute the split terms back into the original polynomial:

$P(a) = 25a^2 - 20a - 15a + 12$

Next, we group the terms into pairs to extract the greatest common monomial from each group:

$P(a) = (25a^2 - 20a) - (15a - 12)$

Factor out the greatest common divisor (GCD) from the first group ($5a$) and the second group ($3$):

$P(a) = 5a(5a - 4) - 3(5a - 4)$

Notice that the binomial $(5a - 4)$ is now a common factor in both terms. We factor out $(5a - 4)$ [by the Distributive Property of Multiplication over Addition]:

$P(a) = (5a - 4)(5a - 3)$

Geometric Visualization

The factorization proves that a rectangle with an area of $25a^2 - 35a + 12$ can be constructed with sides measuring $(5a - 4)$ and $(5a - 3)$. Below is the geometric representation of this relationship.

Final Conclusion

Because multiplication is commutative ($l \times b = b \times l$), either of the two binomial factors can represent the length, and the remaining factor will represent the breadth.

Final Solution: The possible expressions for the length and breadth of the rectangle are $(5a - 3)$ and $(5a - 4)$.

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 $x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8zx$.

Initial Setup & Algebraic Strategy

To evaluate the expression $(998)^3$ without performing direct, cumbersome multiplication, we utilize the fundamental algebraic identities derived from the Binomial Theorem. The objective is to express the base number, $998$, as a binomial $(a - b)$ where $a$ and $b$ are integers that are computationally simple to raise to higher powers.

We can rewrite the base as:

$998 = 1000 - 2$

Thus, the expression becomes:

$(998)^3 = (1000 - 2)^3$

Step 1: Selection of the Appropriate Identity

We apply the standard algebraic identity for the cube of a binomial difference [Per the Binomial Expansion for $n=3$]:

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

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

• $a = 1000$
• $b = 2$

Step 2: Variable Substitution and Expansion

Substituting $a = 1000$ and $b = 2$ into the identity yields:

$(1000 - 2)^3 = (1000)^3 - 3(1000)^2(2) + 3(1000)(2)^2 - (2)^3$

Step 3: Term-by-Term Evaluation

We systematically evaluate each term in the expanded polynomial:

• First Term ($a^3$): $(1000)^3 = 1,000,000,000$
• Second Term ($3a^2b$): $3(1000)^2(2) = 3(1,000,000)(2) = 6,000,000$
• Third Term ($3ab^2$): $3(1000)(2)^2 = 3(1000)(4) = 12,000$
• Fourth Term ($b^3$): $(2)^3 = 8$

Step 4: Arithmetic Synthesis

Now, we substitute the evaluated terms back into the expanded equation and perform the arithmetic operations sequentially to ensure precision:

$(998)^3 = 1,000,000,000 - 6,000,000 + 12,000 - 8$

Operation 1: Subtraction

$1,000,000,000 - 6,000,000 = 994,000,000$

Operation 2: Addition

$994,000,000 + 12,000 = 994,012,000$

Operation 3: Final Subtraction

$994,012,000 - 8 = 994,011,992$

Final Solution: $(998)^3 = 994,011,992$

Step 1: Identify the Applicable Algebraic Identity

To expand the given expression $(\frac{3}{2}x + 1)^3$, we utilize the standard algebraic identity for the cube of a binomial. The expansion of a binomial sum cubed is derived from multiplying the binomial by itself three times: $(a+b)(a+b)(a+b)$.

The standard identity is defined as:

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

[Theoretical Justification: By the Binomial Theorem for $n=3$, the coefficients follow the sequence 1, 3, 3, 1 from Pascal's Triangle, representing the geometric decomposition of a cube into one $a^3$ volume, three $a^2b$ volumes, three $ab^2$ volumes, and one $b^3$ volume.]

Step 2: Map the Variables to the Given Expression

By comparing the given expression $(\frac{3}{2}x + 1)^3$ with the standard identity $(a + b)^3$, we establish the following variable assignments:

• $a = \frac{3}{2}x$
• $b = 1$

Step 3: Term-by-Term Expansion & Algebraic Manipulation

We will now substitute $a$ and $b$ into the identity and rigorously evaluate each of the four terms.

1. First Term ($a^3$):

$ a^3 = \left(\frac{3}{2}x\right)^3 $

$ a^3 = \frac{3^3}{2^3} \cdot x^3 = \frac{27}{8}x^3 $

[Applying the power of a product and quotient rules: $(\frac{p}{q} \cdot x)^n = \frac{p^n}{q^n} \cdot x^n$]

2. Second Term ($3a^2b$):

$ 3a^2b = 3 \cdot \left(\frac{3}{2}x\right)^2 \cdot (1) $

$ 3a^2b = 3 \cdot \left(\frac{9}{4}x^2\right) \cdot 1 $

$ 3a^2b = \frac{27}{4}x^2 $

3. Third Term ($3ab^2$):

$ 3ab^2 = 3 \cdot \left(\frac{3}{2}x\right) \cdot (1)^2 $

$ 3ab^2 = 3 \cdot \left(\frac{3}{2}x\right) \cdot 1 $

$ 3ab^2 = \frac{9}{2}x $

4. Fourth Term ($b^3$):

$ b^3 = (1)^3 = 1 $

Step 4: Synthesize the Final Polynomial

Combine the evaluated terms to form the expanded polynomial. It is standard mathematical convention to write polynomials in descending order of their degree (from the highest power of $x$ to the constant term).

$ \left(\frac{3}{2}x + 1\right)^3 = \frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1 $

Final Solution: The expanded form of the cube is $\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1$.

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 & 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 Expression & Theoretical Foundation

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

$(x + 8)(x - 10)$

[Per the fundamental principles of polynomial multiplication], we can expand this expression using a standard algebraic identity rather than manual term-by-term distribution. This ensures computational efficiency and minimizes arithmetic errors.

Step 1: Selection of the Appropriate Algebraic Identity

The given expression is of the form $(x + a)(x + b)$. The corresponding standard algebraic identity is defined as:

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

Step 2: Variable Mapping

By comparing the given expression $(x + 8)(x - 10)$ with the standard form $(x + a)(x + b)$, we establish the following exact parameter mappings:

• $a = 8$
• $b = -10$

Step 3: Substitution and Expansion

Substituting the mapped values of $a$ and $b$ into the algebraic identity yields:

$(x + 8)(x - 10) = x^2 + (8 + (-10))x + (8)(-10)$

Step 4: Arithmetic Simplification

Next, we simplify the coefficients [By applying the axioms of integer addition and multiplication]:

• The linear coefficient: $8 + (-10) = -2$
• The constant term: $(8)(-10) = -80$

Substituting these simplified values back into the expanded equation gives the final quadratic polynomial:

$x^2 - 2x - 80$

Visual Representation: Area Model of Polynomial Multiplication

The following geometric area model illustrates the partial products of the binomial multiplication. [Note: While geometric lengths cannot physically be negative, the area model serves as a rigorous algebraic abstraction used to visualize the distributive property].

Summing the partial products from the area model confirms our algebraic derivation: $x^2 + 8x - 10x - 80 = x^2 - 2x - 80$.

Final Solution: $x^2 - 2x - 80$

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 $(x - 4)$.