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

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 $\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1$.

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 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 $(2a + b)(2a + b)(2a + b)$.

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.

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

Given Expression & Algebraic Foundation

We are tasked with expanding the following binomial cube:

$(2x + 1)^3$

To expand this expression, we utilize the standard algebraic identity for the cube of a binomial sum. [Per the fundamental theorem of polynomial expansion], the identity is defined as:

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

Step 1: Variable Identification and Substitution

By comparing the given expression $(2x + 1)^3$ with the standard identity $(a + b)^3$, we can establish a direct mapping of the terms:

• Let $a = 2x$
• Let $b = 1$

Substituting these values into the expansion formula yields:

$(2x + 1)^3 = (2x)^3 + 3(2x)^2(1) + 3(2x)(1)^2 + (1)^3$

Step 2: Term-by-Term Simplification

We now systematically simplify each of the four terms generated by the expansion. [Applying the laws of exponents, specifically the power of a product rule: $(xy)^n = x^n y^n$].

• First Term ($a^3$):
  $(2x)^3 = 2^3 \cdot x^3 = 8x^3$


• Second Term ($3a^2b$):
  $3(2x)^2(1) = 3 \cdot (2^2 \cdot x^2) \cdot 1 = 3 \cdot 4x^2 \cdot 1 = 12x^2$


• Third Term ($3ab^2$):
  $3(2x)(1)^2 = 3 \cdot 2x \cdot 1 = 6x$


• Fourth Term ($b^3$):
  $(1)^3 = 1 \cdot 1 \cdot 1 = 1$

Step 3: Final Assembly of the Polynomial

Having simplified each individual component, we combine them sequentially to form the final expanded polynomial in standard form (descending order of degree):

$8x^3 + 12x^2 + 6x + 1$

Final Solution: The expanded form of $(2x + 1)^3$ is $8x^3 + 12x^2 + 6x + 1$.

Initial Setup & Algebraic Identity Selection

We are tasked with evaluating the expression $(102)^3$ without performing direct multiplication. To achieve this, we utilize standard algebraic identities by expressing the base number as a binomial.

Step 1: Decomposition of the Base Number

To simplify the calculation, we decompose the base number $102$ into the sum of two numbers. It is computationally optimal to choose a multiple of $10$ as the primary term.

$102 = 100 + 2$

Therefore, the expression can be rewritten as:

$(102)^3 = (100 + 2)^3$

Step 2: Application of the Cubic Identity

[Per the standard binomial expansion for cubes], the relevant algebraic identity is:

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

Alternatively, the fully expanded form of this identity is:

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

By mapping our decomposed values to the variables in the identity, we establish:

• $a = 100$
• $b = 2$

Step 3: Substitution and Expansion

Substituting $a = 100$ and $b = 2$ into the factored form of the identity yields:

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

Step 4: Computation of Individual Terms

We now calculate each term systematically, adhering to the order of operations:

• Cube of the first term: $(100)^3 = 100 \times 100 \times 100 = 1,000,000$
• Cube of the second term: $(2)^3 = 2 \times 2 \times 2 = 8$
• Product of the variables: $3ab = 3(100)(2) = 600$
• Multiplication of the product term with the binomial sum: $600(100 + 2) = 600(102) = 61,200$

Note: The final term can also be computed via distribution: $600(100) + 600(2) = 60,000 + 1,200 = 61,200$.

Step 5: Final Summation

We combine the computed values to find the final result:

$1,000,000 + 8 + 61,200$

Aligning the values for addition:

$\phantom{+}1,000,000$
$\phantom{+00}61,200$
$\underline{+\phantom{000000}8}$
$\phantom{+}1,061,208$

Final Solution: The evaluated value of $(102)^3$ is 1,061,208.

Given Expression & Theoretical Foundation

We are tasked with expanding the binomial cube: $(2a - 3b)^3$.

To expand a binomial raised to the third power, we utilize the standard algebraic identity for the cube of a difference. [Per the fundamental theorems of polynomial expansion], the identity is defined as:

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

Alternatively, it can be factored as $(x - y)^3 = x^3 - y^3 - 3xy(x - y)$. For standard polynomial expansion, the fully distributed form is preferred.

Step 1: Identification and Mapping of Variables

By comparing our given expression $(2a - 3b)^3$ with the standard identity $(x - y)^3$, we map the variables as follows:

• $x = 2a$
• $y = 3b$

Step 2: Substitution into the Algebraic Identity

Substitute $x = 2a$ and $y = 3b$ into the expanded form of the identity:

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

Step 3: Expansion and Simplification of Individual Terms

We will now apply the laws of exponents [specifically $(uv)^n = u^n v^n$] to evaluate each term systematically:

• First Term ($x^3$):
  $(2a)^3 = 2^3 \cdot a^3 = 8a^3$


• Second Term ($-3x^2y$):
  $-3(2a)^2(3b) = -3(4a^2)(3b)$
  Multiply the numerical coefficients: $-3 \cdot 4 \cdot 3 = -36$
  Multiply the variables: $a^2 \cdot b = a^2b$
  Result: $-36a^2b$


• Third Term ($+3xy^2$):
  $3(2a)(3b)^2 = 3(2a)(9b^2)$
  Multiply the numerical coefficients: $3 \cdot 2 \cdot 9 = 54$
  Multiply the variables: $a \cdot b^2 = ab^2$
  Result: $+54ab^2$


• Fourth Term ($-y^3$):
  $-(3b)^3 = -(3^3 \cdot b^3) = -27b^3$

Step 4: Final Assembly of the Polynomial

Combine all the simplified terms from Step 3 in descending order of the degree of $a$:

$(2a - 3b)^3 = 8a^3 - 36a^2b + 54ab^2 - 27b^3$

Final Solution: The expanded form of the cube $(2a - 3b)^3$ is $8a^3 - 36a^2b + 54ab^2 - 27b^3$.

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 $(x - 4)$.

Initial Setup & Given Expression

We are tasked with finding the product of the following binomials using a suitable algebraic identity:

$(3x + 4)(3x - 5)$

Step 1: Identifying the Suitable Algebraic Identity

Observing the given expression, both binomials share the exact same first term ($3x$) but possess different second terms ($+4$ and $-5$). Therefore, the most appropriate standard algebraic identity to apply is:

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

[Per the fundamental algebraic identity for the product of two binomials with a common term, derived from the distributive property of multiplication over addition].

Step 2: Variable Mapping

By comparing our given expression $(3x + 4)(3x - 5)$ with the standard identity $(X + a)(X + b)$, we can establish the following precise variable mappings:

• $X = 3x$
• $a = 4$
• $b = -5$

Step 3: Substitution and Expansion

Substituting these mapped values into the right-hand side of the identity yields:

$(3x + 4)(3x - 5) = (3x)^2 + (4 + (-5))(3x) + (4)(-5)$

Step 4: Simplification of Terms

We now simplify each term systematically to construct the final polynomial:

• Quadratic Term: $(3x)^2 = 3^2 \cdot x^2 = 9x^2$
  [Applying the power of a product rule: $(xy)^n = x^n y^n$]
• Linear Term: $(4 - 5)(3x) = (-1)(3x) = -3x$
  [Combining constant coefficients algebraically and multiplying by the common variable term]
• Constant Term: $(4)(-5) = -20$
  [The product of a positive integer and a negative integer is negative]

Combining these simplified terms yields the final quadratic polynomial:

$9x^2 - 3x - 20$

Visual Representation: The Tabular Method (Area Model)

The following grid visually demonstrates the distributive property (often referred to as the FOIL method), confirming our algebraic expansion by calculating the partial products of each individual term.

Summing the partial products from the interior of the grid gives:

$9x^2 + 12x - 15x - 20$

Combining the like linear terms ($+12x$ and $-15x$) results in $-3x$, which perfectly matches our identity-based calculation: $9x^2 - 3x - 20$. [This verifies our result via the distributive property of multiplication over addition].

Final Solution: The product of $(3x + 4)(3x - 5)$ is $9x^2 - 3x - 20$.

Initial Setup & Algebraic Framework

We are tasked with finding the product of two binomials: $(x + 4)(x + 10)$.

To expand this expression efficiently, we analyze the structure of the binomials. Both binomials share a common first term ($x$) and have distinct constant terms ($4$ and $10$). This structure perfectly matches the standard algebraic identity for the product of two binomials with a common term.

Step 1: Identifying the Suitable Identity

According to the fundamental principles of polynomial expansion [derived via the distributive property of multiplication over addition], the product of two binomials of the form $(x + a)$ and $(x + b)$ is given by the identity:

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

Step 2: Mapping the Variables

By comparing the given expression $(x + 4)(x + 10)$ with the standard identity $(x + a)(x + b)$, we can establish a direct correspondence for the constants:

• $a = 4$
• $b = 10$

Step 3: Substitution and Evaluation

Substituting the identified values of $a$ and $b$ into the algebraic identity, we proceed with the expansion:

$(x + 4)(x + 10) = x^2 + (4 + 10)x + (4)(10)$

Now, we perform the arithmetic operations within the parentheses [combining like terms and computing the product of the constants]:

• Sum of the constants: $4 + 10 = 14$
• Product of the constants: $4 \times 10 = 40$

Substituting these results back into the equation yields the expanded quadratic polynomial:

$(x + 4)(x + 10) = x^2 + 14x + 40$

Step 4: Geometric Interpretation (Area Model)

The algebraic expansion can be rigorously verified using a geometric area model. The product $(x + 4)(x + 10)$ represents the total area of a rectangle with dimensions $(x + 4)$ and $(x + 10)$. By partitioning this rectangle into four smaller rectangular regions, the sum of their individual areas equals the total expanded polynomial.

As demonstrated in the geometric model, the total area is the sum of the four sub-areas: $x^2 + 10x + 4x + 40$. Combining the linear terms ($10x$ and $4x$) simplifies the expression precisely to $x^2 + 14x + 40$.

Final Solution: $(x + 4)(x + 10) = x^2 + 14x + 40$

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 $

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 & 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: $x^2 - 2x - 80$

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 & Selection of Algebraic Identity

To evaluate the product $95 \times 96$ without performing direct multiplication, we must utilize a standard algebraic identity. The most efficient approach is to express both numbers as binomials relative to a common, easily calculable base. The number $100$ serves as the optimal base because squaring it and multiplying it by other integers minimizes computational complexity.

We will apply the following binomial product identity:

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

[Per the distributive property of multiplication over addition, expanding the binomials yields $x^2 + bx + ax + ab$, which factors into the stated identity].

Step 1: Expressing the Factors as Binomials

We rewrite the given numbers, $95$ and $96$, in terms of the base $x = 100$:

• $95 = 100 - 5$
• $96 = 100 - 4$

Thus, the product $95 \times 96$ can be written as $(100 - 5)(100 - 4)$.

Step 2: Mapping Variables to the Identity

By comparing $(100 - 5)(100 - 4)$ with the standard identity $(x + a)(x + b)$, we establish the following variable assignments:

• $x = 100$
• $a = -5$
• $b = -4$

Step 3: Substituting and Expanding

Substitute the assigned values into the right-hand side of the identity $x^2 + (a + b)x + ab$:

$(100 - 5)(100 - 4) = (100)^2 + (-5 + -4)(100) + (-5)(-4)$

Step 4: Arithmetic Evaluation

Now, we evaluate each term systematically [adhering to the order of operations, PEMDAS/BODMAS]:

• First term (Square of the base): $(100)^2 = 10000$
• Second term (Sum of constants multiplied by the base): $(-5 - 4) \times 100 = (-9) \times 100 = -900$
• Third term (Product of the constants): $(-5) \times (-4) = 20$ [Per the rule that the product of two negative integers yields a positive integer]

Step 5: Final Aggregation

Combine the evaluated terms to find the final product:

$10000 - 900 + 20$

First, subtract $900$ from $10000$:

$10000 - 900 = 9100$

Next, add $20$ to the result:

$9100 + 20 = 9120$

Final Solution: 9120

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.

Initial Setup & Algebraic Identity

We are tasked with expanding the algebraic expression:

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

To expand the cube of a binomial difference, we utilize the standard algebraic identity derived from the Binomial Theorem [where $n=3$]. The identity for the cube of a difference is given by:

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

Step 1: Variable Mapping and Substitution

By comparing our given expression $(x - \frac{2}{3}y)^3$ with the standard identity $(a - b)^3$, we can establish the following one-to-one mapping of variables:

• $a = x$
• $b = \frac{2}{3}y$

Substituting these mapped values into the expansion formula yields:

$(x - \frac{2}{3}y)^3 = (x)^3 - 3(x)^2(\frac{2}{3}y) + 3(x)(\frac{2}{3}y)^2 - (\frac{2}{3}y)^3$

Step 2: Term-by-Term Expansion and Simplification

We will now isolate and simplify each of the four terms generated by the expansion [adhering strictly to the order of operations and the laws of exponents, specifically $(xy)^n = x^n y^n$].

• First Term ($a^3$):
  $(x)^3 = x^3$


• Second Term ($-3a^2b$):
  $-3(x)^2(\frac{2}{3}y) = -3 \cdot x^2 \cdot \frac{2}{3}y$
  The scalar $3$ in the numerator and the $3$ in the denominator cancel out:
  $= -2x^2y$


• Third Term ($+3ab^2$):
  $3(x)(\frac{2}{3}y)^2$
  First, square the term inside the parentheses [$(\frac{2}{3})^2 \cdot y^2 = \frac{4}{9}y^2$]:
  $= 3 \cdot x \cdot \frac{4}{9}y^2$
  Multiply the scalars ($3 \cdot \frac{4}{9} = \frac{12}{9}$), and reduce the fraction by dividing the numerator and denominator by their greatest common divisor, $3$:
  $= \frac{4}{3}xy^2$


• Fourth Term ($-b^3$):
  $-(\frac{2}{3}y)^3$
  Cube both the coefficient and the variable [$(\frac{2}{3})^3 = \frac{2^3}{3^3} = \frac{8}{27}$]:
  $= -\frac{8}{27}y^3$

Step 3: Final Assembly of the Polynomial

Combine the simplified terms from Step 2 in descending order of the degree of $x$ to form the final expanded polynomial:

$x^3 - 2x^2y + \frac{4}{3}xy^2 - \frac{8}{27}y^3$

Final Solution: The expanded form of $(x - \frac{2}{3}y)^3$ is $x^3 - 2x^2y + \frac{4}{3}xy^2 - \frac{8}{27}y^3$.