Polynomials and Factoring

Learning Objectives

• Solve equations involving polynomial addition, subtraction, multiplication, and special products.

• Factor algebraic expressions using the greatest common factor (GCF), grouping, and trinomial patterns.

Definition and Structure of Polynomials

• A monomial is an algebraic expression consisting of a single term (a constant multiplied by variables with non-negative integer exponents).

• A binomial is a sum of two monomials.

• A trinomial is a sum of three monomials.

• A polynomial in $x$ is a sum of any number of monomials in $x$.

Definition of Polynomial

• A polynomial in $x$ is a sum of the form:   $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$   where $n$ is a non-negative integer and each coefficient $a_k$ is a real number.

• If $a_n \neq 0$, then the polynomial is said to have degree $n$.

• The coefficient $a_n$ is called the leading coefficient.

Equality of Polynomials

• By definition, two polynomials are equal if and only if:

  • They have the same degree.

  • The coefficients of like powers of $x$ are equal.

Examples of Polynomials

Constant and Zero Polynomials

• Zero Polynomial: If all coefficients of a polynomial are zero, it is called the zero polynomial and is denoted by $0$.

  • By convention, the degree of the zero polynomial is undefined (it is not zero).

• Constant Polynomials: If $c$ is a non-zero real number, then $c$ is a polynomial of degree 0$.\n * Constant polynomials consist of non-zero real constants together with the zero polynomial.\n\n## Formatting and Conventions\n\n* **Negative Coefficients**: When a coefficient is negative, a minus sign is usually written between terms rather than keeping explicit addition of negative terms.\n * Example: 3x^2 + (-5)x + (-7) = 3x^2 - 5x - 7\n* **Polynomials in Other Variables**: Polynomials can be defined using variables other than x$.

  • Example: $z^2 - 3z^7 + 8 - \sqrt{5}z^4$ is a polynomial in $z$ of degree 7$.\n* **Decreasing Order of Powers (Standard Form)**: Polynomial terms are typically arranged in order of decreasing powers of the variable.\n * Example: Rearranging z^2 - 3z^7 + 8 - \sqrt{5}z^4$yields$-3z^7 - \sqrt{5}z^4 + z^2 + 8\n\n## Nonpolynomial Algebraic Expressions\n\n* A polynomial in x$is an algebraic expression formed using a finite number of additions, subtractions, and multiplications involving$x$.

• If an expression involves division by a variable or roots containing a variable, it is not a polynomial.

• Nonpolynomial Examples:

  • $\frac{1}{x} + 3x$ (contains division by the variable $x$)

  • $\frac{x - 5}{x^2 + 2}$ (contains division by an expression containing $x$)

  • $3x^2 + \sqrt{x} - 2$ (contains a radical with the variable $x$ inside)

Polynomial Operations

Addition and Subtraction

• Addition: To add two polynomials in $x$, combine like terms by adding the coefficients of like powers of x$.\n* **Subtraction**: To subtract polynomials, remove parentheses by changing the sign of every term in the subtracted polynomial, then combine like terms.\n\n### Example 1: Adding and Subtracting Polynomials\n\n* **(a) Sum**: (x^3 + 2x^2 - 5x + 7) + (4x^3 - 5x^2 + 3)\n * Step 1 (Remove parentheses): x^3 + 2x^2 - 5x + 7 + 4x^3 - 5x^2 + 3\n * Step 2 (Group and add coefficients of like powers): (1 + 4)x^3 + (2 - 5)x^2 - 5x + (7 + 3)\n * Step 3 (Simplify): 5x^3 - 3x^2 - 5x + 10\n* **(b) Difference**: (x^3 + 2x^2 - 5x + 7) - (4x^3 - 5x^2 + 3)\n * Step 1 (Remove parentheses and change signs): x^3 + 2x^2 - 5x + 7 - 4x^3 + 5x^2 - 3\n * Step 2 (Group and add coefficients of like powers): (1 - 4)x^3 + (2 + 5)x^2 - 5x + (7 - 3)\n * Step 3 (Simplify): -3x^3 + 7x^2 - 5x + 4\n\n## Multiplication of Polynomials\n\n* Multiply every term of the first polynomial by every term of the second polynomial using distributive properties, then combine like terms.\n\n### Example 2: Multiplying Binomials\n\n* Find the product: (4x + 5)(3x - 2)\n * Rewrite subtraction as addition: 3x - 2 = 3x + (-2)\n * Apply distributive properties: (4x)(3x) + (4x)(-2) + (5)(3x) + (5)(-2)\n * Multiply individual terms: 12x^2 - 8x + 15x - 10\n * Simplify by combining like terms: 12x^2 + 7x - 10\n\n### Example 3: Multiplying Polynomials\n\n* Find the product: (x^2 + 5x - 4)(2x^3 + 3x - 1)\n* **Method 1 (Horizontal Distributive Property)**:\n * Treat (2x^3 + 3x - 1) as a single expression:\n    x^2(2x^3 + 3x - 1) + 5x(2x^3 + 3x - 1) - 4(2x^3 + 3x - 1)\n * Apply distributive property across each term:\n    2x^5 + 3x^3 - x^2 + 10x^4 + 15x^2 - 5x - 8x^3 - 12x + 4\n * Combine like terms (yielding 9 initial products reduced to standard form):\n    2x^5 + 10x^4 - 5x^3 + 14x^2 - 17x + 4\n* **Method 2 (Vertical Multiplication)**:\n * Align polynomials vertically, leaving blank spaces for missing powers of x (terms with zero coefficients):\n\n\n\n * Step-by-step product lines:\n * Line 1 (x^2 \times \text{top}$):$2x^5 + 3x^3 - x^2\n * Line 2 (5x \times \text{top}$):$10x^4 + 15x^2 - 5x\n * Line 3 (-4 \times \text{top}$):$-8x^3 - 12x + 4\n * Sum of lines: 2x^5 + 10x^4 - 5x^3 + 14x^2 - 17x + 4\n\n## Division of a Polynomial by a Monomial\n\n* To divide a polynomial by a monomial, divide each individual term of the polynomial numerator by the monomial denominator, then simplify exponents.\n\n### Example 4: Dividing a Polynomial by a Monomial\n\n* Express as a polynomial in x$and$y:\n  \frac{6x^2 y^3 + 4x^3 y^2 - 10xy}{2xy}\n* Divide each term individually:\n  \frac{6x^2 y^3}{2xy} + \frac{4x^3 y^2}{2xy} - \frac{10xy}{2xy}\n* Simplify terms:\n  3xy^2 + 2x^2 y - 5\n\n# Product Formulas\n\n## Special Product Identity Table\n\n| Identity Formula | Specific Illustration |\n| :--- | :--- |\n| **(1) Difference of Squares**:
  (x + y)(x - y) = x^2 - y^2$|$(2a + 3)(2a - 3) = (2a)^2 - 3^2 = 4a^2 - 9 |\n| **(2) Perfect Squares**:
  (x \pm y)^2 = x^2 \pm 2xy + y^2$|$(2a - 3)^2 = (2a)^2 - 2(2a)(3) + 3^2 = 4a^2 - 12a + 9 |\n| **(3) Cube of a Binomial**:
  (x \pm y)^3 = x^3 \pm 3x^2 y + 3xy^2 \pm y^3$|$(2a + 3)^3 = (2a)^3 + 3(2a)^2(3) + 3(2a)(3)^2 + 3^3 = 8a^3 + 36a^2 + 54a + 27 |\n\n## Example 5: Using Product Formulas\n\n* **(a) Product of Sum and Difference**:\n * Problem: (2r^2 - \sqrt{s})(2r^2 + \sqrt{s})\n * Solution: Apply Product Formula 1 with x = 2r^2$and$y = \sqrt{s}:\n    (2r^2 - \sqrt{s})(2r^2 + \sqrt{s}) = (2r^2)^2 - (\sqrt{s})^2 = 4r^4 - s\n* **(b) Square of a Binomial with Radicals**:\n * Problem: \left(\sqrt{c} + \frac{1}{\sqrt{c}}\right)^2\n * Solution: Apply Product Formula 2 with x = \sqrt{c}$and$y = \frac{1}{\sqrt{c}}:\n    \left(\sqrt{c} + \frac{1}{\sqrt{c}}\right)^2 = (\sqrt{c})^2 + 2 \cdot \sqrt{c} \cdot \frac{1}{\sqrt{c}} + \left(\frac{1}{\sqrt{c}}\right)^2 = c + 2 + \frac{1}{c}\n * Note: The resulting expression c + 2 + \frac{1}{c}$is **not** a polynomial due to the term$\frac{1}{c}.\n* **(c) Cube of a Binomial Difference**:\n * Problem: (2a - 5b)^3\n * Solution: Apply Product Formula 3 with x = 2a$and$y = 5b:\n    (2a - 5b)^3 = (2a)^3 - 3(2a)^2(5b) + 3(2a)(5b)^2 - (5b)^3 = 8a^3 - 60a^2 b + 150ab^2 - 125b^3\n\n# Factoring Polynomials\n\n## Fundamentals of Factoring\n\n* **Factor**: If a polynomial is written as a product of other polynomials, each polynomial in that product is a factor of the original polynomial.\n* **Factoring**: The process of expressing a sum of terms as a product.\n * Example: Since x^2 - 9 = (x + 3)(x - 3)$, the expressions$(x + 3)$and$(x - 3)$are factors of$x^2 - 9$.

• Purpose: Factoring simplifies mathematical analysis by reducing a complex expression into several simpler component expressions.

• Nontrivial Factors: Factors containing polynomials of positive degree.

• Integral Factors: When restricting coefficients to integers, common integer factors are factored out from all terms.

  • Example: $4x^2 y + 8z^3 = 4(x^2 y + 2z^3)$

• Irreducible Polynomial: A polynomial that cannot be factored into polynomials of lower positive degree.

• To Factor a Polynomial: Expressing it completely as a product of irreducible polynomials.

• Greatest Common Factor (GCF): The product of all common factors present in every term, with each factor raised to the smallest nonzero exponent occurring in any single term.

Factoring Common Factors (GCF Examples)

• Always factor out the GCF first before applying higher-level factoring formulas:

  • $8x^2 + 4xy = 4x(2x + y)$

  • $25x^2 + 25x - 150 = 25(x^2 + x - 6) = 25(x + 3)(x - 2)$

  • $4x^5 y - 9x^3 y^3 = x^3 y(4x^2 - 9y^2) = x^3 y(2x + 3y)(2x - 3y)$

Factoring Formulas Table

Example 6: Difference of Two Squares

• (a) Factor $25r^2 - 49s^2$:

  • Apply difference of two squares with $x = 5r$ and $y = 7s$:     $25r^2 - 49s^2 = (5r)^2 - (7s)^2 = (5r + 7s)(5r - 7s)$

• (b) Factor $81x^4 - y^4$:

  • Write $81x^4 = (9x^2)^2$ and $y^4 = (y^2)^2$:     $81x^4 - y^4 = (9x^2)^2 - (y^2)^2 = (9x^2 + y^2)(9x^2 - y^2)$

  • Apply the formula a second time to the factor $(9x^2 - y^2)$:     $= (9x^2 + y^2)[(3x)^2 - y^2] = (9x^2 + y^2)(3x + y)(3x - y)$

• (c) Factor $16x^4 - (y - 2z)^2$:

  • Write $16x^4 = (4x^2)^2$:     $16x^4 - (y - 2z)^2 = (4x^2)^2 - (y - 2z)^2$

  • Apply difference of two squares:     $= [(4x^2) + (y - 2z)][(4x^2) - (y - 2z)] = (4x^2 + y - 2z)(4x^2 - y + 2z)$

Example 7: Sum and Difference of Two Cubes

• (a) Factor $a^3 + 64b^3$:

  • Apply sum of two cubes formula with $x = a$ and $y = 4b$:     $a^3 + 64b^3 = a^3 + (4b)^3 = (a + 4b)[a^2 - a(4b) + (4b)^2] = (a + 4b)(a^2 - 4ab + 16b^2)$

• (b) Factor $8c^6 - 27d^9$:

  • Apply difference of two cubes formula with $x = 2c^2$ and $y = 3d^3$:     $8c^6 - 27d^9 = (2c^2)^3 - (3d^3)^3 = (2c^2 - 3d^3)[(2c^2)^2 + (2c^2)(3d^3) + (3d^3)^2]$     $= (2c^2 - 3d^3)(4c^4 + 6c^2 d^3 + 9d^6)$

Factoring Trinomials by Trial and Error

• A trinomial $px^2 + qx + r$ with integer coefficients $p, q, r$ factors into:   $px^2 + qx + r = (ax + b)(cx + d)$   where $a, b, c, d$ are integers satisfying:   $ac = p$   $bd = r$   $ad + bc = q$

• If no combination of integer values for $a, b, c, d$ satisfies these conditions, the trinomial $px^2 + qx + r$ is irreducible.

• This method extends to trinomials of two variables $px^2 + qxy + ry^2 = (ax + by)(cx + dy)$.

Example 8: Factoring a Trinomial by Trial and Error

• Factor $6x^2 - 7x - 3$:

  • Required conditions for $6x^2 - 7x - 3 = (ax + b)(cx + d)$:     $ac = 6$     $bd = -3$     $ad + bc = -7$

  • Assuming positive values for $a$ and $c$, the candidate pairs for $(a, c)$ are:     | $a$ | $1$ | $6$ | $2$ | $3$ |     | :--- | :--- | :--- | :--- | :--- |     | $c$ | $6$ | $1$ | $3$ | $2$ |

  • This gives four potential bracket forms:

  1. $(x + b)(6x + d)$

  2. $(6x + b)(x + d)$

  3. $(2x + b)(3x + d)$

  4. $(3x + b)(2x + d)$

  • Candidate pairs for $(b, d)$ satisfying $bd = -3$ are:     | $b$ | $1$ | $-1$ | $3$ | $-3$ |     | :--- | :--- | :--- | :--- | :--- |     | $d$ | $-3$ | $3$ | $-1$ | $1$ |

  • Testing combinations for $ad + bc = -7$:

  • Testing $(2x + b)(3x + d)$ with $b = -3$ and $d = 1$:       $ad + bc = (2)(1) + (-3)(3) = 2 - 9 = -7$

  • Correct Factorization:     $6x^2 - 7x - 3 = (2x - 3)(3x + 1)$

Factoring by Grouping

• For expressions with four or more terms, group terms into pairs or subsets to factor out common binomial expressions.

Example 10: Factoring by Grouping

• (a) Factor $4ac + 2bc - 2ad - bd$:

  • Group first two terms and last two terms:     $(4ac + 2bc) - (2ad + bd)$

  • Factor out common monomial from each pair:     $= 2c(2a + b) - d(2a + b)$

  • Factor out common binomial $(2a + b)$:     $= (2c - d)(2a + b) \quad \text{or} \quad (2a + b)(2c - d)$

• (b) Factor $3x^3 + 2x^2 - 12x - 8$:

  • Group terms:     $(3x^3 + 2x^2) - (12x + 8)$

  • Factor out common factors:     $= x^2(3x + 2) - 4(3x + 2)$

  • Factor out $(3x + 2)$:     $= (x^2 - 4)(3x + 2)$

  • Factor $(x^2 - 4)$ using difference of two squares:     $= (x + 2)(x - 2)(3x + 2)$

• (c) Factor $x^2 - 16y^2 + 10x + 25$:

  • Rearrange and group as a perfect square trinomial minus a square term:     $(x^2 + 10x + 25) - 16y^2$

  • Rewrite grouped terms as perfect squares:     $= (x + 5)^2 - (4y)^2$

  • Apply difference of two squares with $x \rightarrow (x + 5)$ and $y \rightarrow 4y$:     $= [(x + 5) - 4y][(x + 5) + 4y]$

  • Simplify inside brackets:     $= (x - 4y + 5)(x + 4y + 5)$

Bibliographic Reference

• Title: College Algebra, 8th Edition

• Authors: James Stewart, Lothar Redlin, Saleem Watson

• Publication Year: 2024

• ISBN: 9780357753651

Learning Objectives

The primary learning objectives are to solve equations involving polynomial addition, subtraction, multiplication, and special products, as well as to factor algebraic expressions using the greatest common factor (GCF), grouping, and trinomial patterns.

Definition and Structure of Polynomials

A monomial is an algebraic expression consisting of a single term, which is a constant multiplied by variables with non-negative integer exponents. A binomial is a sum of two monomials, while a trinomial is a sum of three monomials. In general, a polynomial in $x$ is a sum of any number of monomials in $x$.

Definition of Polynomial

Formally, a polynomial in $x$ is a sum of the form $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer and each coefficient $a_k$ is a real number. If $a_n \neq 0$, then the polynomial is said to have degree $n$, and the coefficient $a_n$ is called the leading coefficient.

Equality of Polynomials

By definition, two polynomials are equal if and only if they have the same degree and the coefficients of like powers of $x$ are equal.

Constant and Zero Polynomials

If all coefficients of a polynomial are zero, it is called the zero polynomial and is denoted by $0$. By convention, the degree of the zero polynomial is undefined rather than zero. Conversely, if $c$ is a non-zero real number, then $c$ is a polynomial of degree $0$. Constant polynomials consist of non-zero real constants together with the zero polynomial.

Formatting and Conventions

When a coefficient is negative, a minus sign is usually written between terms rather than keeping explicit addition of negative terms, such as writing $3x^2 + (-5)x + (-7)$ as $3x^2 - 5x - 7$. Polynomials can also be defined using variables other than $x$; for instance, $z^2 - 3z^7 + 8 - \sqrt{5}z^4$ is a polynomial in $z$ of degree $7$. Polynomial terms are typically arranged in standard form, which is in order of decreasing powers of the variable. Rearranging $z^2 - 3z^7 + 8 - \sqrt{5}z^4$ yields $-3z^7 - \sqrt{5}z^4 + z^2 + 8$.

Nonpolynomial Algebraic Expressions

A polynomial in $x$ is an algebraic expression formed using a finite number of additions, subtractions, and multiplications involving $x$. If an expression involves division by a variable or roots containing a variable, it is not a polynomial. Examples of nonpolynomial expressions include $\frac{1}{x} + 3x$ because it contains division by the variable $x$, $\frac{x - 5}{x^2 + 2}$ because it contains division by an expression containing $x$, and $3x^2 + \sqrt{x} - 2$ because it contains a radical with the variable $x$ inside.

Polynomial Operations

Addition and Subtraction

To add two polynomials in $x$, combine like terms by adding the coefficients of like powers of $x$. To subtract polynomials, remove parentheses by changing the sign of every term in the subtracted polynomial, and then combine like terms.

For example, to find the sum $(x^3 + 2x^2 - 5x + 7) + (4x^3 - 5x^2 + 3)$, remove parentheses to get $x^3 + 2x^2 - 5x + 7 + 4x^3 - 5x^2 + 3$, group coefficients of like powers as $(1 + 4)x^3 + (2 - 5)x^2 - 5x + (7 + 3)$, and simplify to $5x^3 - 3x^2 - 5x + 10$. To find the difference $(x^3 + 2x^2 - 5x + 7) - (4x^3 - 5x^2 + 3)$, change the signs of the second polynomial to get $x^3 + 2x^2 - 5x + 7 - 4x^3 + 5x^2 - 3$, group coefficients as $(1 - 4)x^3 + (2 + 5)x^2 - 5x + (7 - 3)$, and simplify to -3x^3 + 7x^2 - 5x + 4$.\n\n#### Multiplication of Polynomials\n\nTo multiply polynomials, multiply every term of the first polynomial by every term of the second polynomial using distributive properties, and then combine like terms.\n\nFor instance, multiplying binomials (4x + 5)(3x - 2)$involves applying the distributive property to obtain$(4x)(3x) + (4x)(-2) + (5)(3x) + (5)(-2)$, which expands to$12x^2 - 8x + 15x - 10$and simplifies to$12x^2 + 7x - 10.\n\nWhen multiplying higher degree polynomials such as (x^2 + 5x - 4)(2x^3 + 3x - 1)$, one can use the horizontal distributive property by expanding$x^2(2x^3 + 3x - 1) + 5x(2x^3 + 3x - 1) - 4(2x^3 + 3x - 1)$into$2x^5 + 3x^3 - x^2 + 10x^4 + 15x^2 - 5x - 8x^3 - 12x + 4$, which simplifies to$2x^5 + 10x^4 - 5x^3 + 14x^2 - 17x + 4$. Alternatively, vertical multiplication can be used by listing polynomials vertically with blank spaces for missing powers of$x$. Line 1 ($x^2$times top) is$2x^5 + 3x^3 - x^2$, line 2 ($5x$times top) is$10x^4 + 15x^2 - 5x$, and line 3 ($-4$times top) is$-8x^3 - 12x + 4$, which sum to$2x^5 + 10x^4 - 5x^3 + 14x^2 - 17x + 4$.

Division of a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each individual term of the polynomial numerator by the monomial denominator, then simplify exponents. For example, dividing $\frac{6x^2 y^3 + 4x^3 y^2 - 10xy}{2xy}$ into $\frac{6x^2 y^3}{2xy} + \frac{4x^3 y^2}{2xy} - \frac{10xy}{2xy}$ yields $3xy^2 + 2x^2 y - 5$.

Special Product Formulas

Special product formulas streamline polynomial multiplication. The difference of squares formula is $(x + y)(x - y) = x^2 - y^2$, illustrated by $(2a + 3)(2a - 3) = (2a)^2 - 3^2 = 4a^2 - 9$. The perfect squares formula is $(x \pm y)^2 = x^2 \pm 2xy + y^2$, illustrated by $(2a - 3)^2 = (2a)^2 - 2(2a)(3) + 3^2 = 4a^2 - 12a + 9$. The cube of a binomial formula is $(x \pm y)^3 = x^3 \pm 3x^2 y + 3xy^2 \pm y^3$, illustrated by (2a + 3)^3 = (2a)^3 + 3(2a)^2(3) + 3(2a)(3)^2 + 3^3 = 8a^3 + 36a^2 + 54a + 27$.\n\nTo compute the product of a sum and difference like (2r^2 - \sqrt{s})(2r^2 + \sqrt{s})$, apply the difference of squares formula with$x = 2r^2$and$y = \sqrt{s}$to obtain$(2r^2)^2 - (\sqrt{s})^2 = 4r^4 - s$.

To square a binomial with radicals like $\left(\sqrt{c} + \frac{1}{\sqrt{c}}\right)^2$, apply the perfect square formula with $x = \sqrt{c}$ and $y = \frac{1}{\sqrt{c}}$ to get $(\sqrt{c})^2 + 2 \cdot \sqrt{c} \cdot \frac{1}{\sqrt{c}} + \left(\frac{1}{\sqrt{c}}\right)^2 = c + 2 + \frac{1}{c}$. Note that the resulting expression $c + 2 + \frac{1}{c}$ is not a polynomial due to the term $\frac{1}{c}$.

To expand the cube of a binomial difference such as $(2a - 5b)^3$, apply the cube formula with $x = 2a$ and $y = 5b$ to obtain (2a)^3 - 3(2a)^2(5b) + 3(2a)(5b)^2 - (5b)^3 = 8a^3 - 60a^2 b + 150ab^2 - 125b^3$.\n\n### Factoring Polynomials\n\n#### Fundamentals of Factoring\n\nIf a polynomial is written as a product of other polynomials, each polynomial in that product is a factor of the original polynomial. Factoring is the process of expressing a sum of terms as a product; for example, since x^2 - 9 = (x + 3)(x - 3)$, the expressions$(x + 3)$and$(x - 3)$are factors of$x^2 - 9$. Factoring simplifies mathematical analysis by reducing a complex expression into several simpler component expressions. Nontrivial factors are factors containing polynomials of positive degree. When restricting coefficients to integers, common integer factors are factored out from all terms, such as$4x^2 y + 8z^3 = 4(x^2 y + 2z^3). An irreducible polynomial is a polynomial that cannot be factored into polynomials of lower positive degree. Factoring a polynomial completely means expressing it as a product of irreducible polynomials. The greatest common factor (GCF) is the product of all common factors present in every term, with each factor raised to the smallest nonzero exponent occurring in any single term.\n\n#### Factoring Common Factors\n\nAlways factor out the GCF first before applying higher-level factoring formulas. For example, factoring 8x^2 + 4xy$yields$4x(2x + y)$. For$25x^2 + 25x - 150$, factor out the GCF$25$to get$25(x^2 + x - 6)$, which further factors into$25(x + 3)(x - 2)$. For$4x^5 y - 9x^3 y^3$, factor out$x^3 y$to get$x^3 y(4x^2 - 9y^2)$, which factors into$x^3 y(2x + 3y)(2x - 3y).\n\n#### Difference of Two Squares\n\nThe difference of two squares pattern can be applied to factor expressions into conjugate pairs. For example, factoring 25r^2 - 49s^2$with$x = 5r$and$y = 7s$yields$(5r)^2 - (7s)^2 = (5r + 7s)(5r - 7s)$. When factoring$81x^4 - y^4$, write the terms as$(9x^2)^2 - (y^2)^2 = (9x^2 + y^2)(9x^2 - y^2)$, and then apply the formula a second time to$(9x^2 - y^2)$to get$(9x^2 + y^2)(3x + y)(3x - y)$. Similarly, factoring$16x^4 - (y - 2z)^2$as$(4x^2)^2 - (y - 2z)^2$results in$[(4x^2) + (y - 2z)][(4x^2) - (y - 2z)] = (4x^2 + y - 2z)(4x^2 - y + 2z).\n\n#### Sum and Difference of Two Cubes\n\nTo factor a^3 + 64b^3$, apply the sum of two cubes formula with$x = a$and$y = 4b$to get$a^3 + (4b)^3 = (a + 4b)[a^2 - a(4b) + (4b)^2] = (a + 4b)(a^2 - 4ab + 16b^2)$. To factor$8c^6 - 27d^9$, apply the difference of two cubes formula with$x = 2c^2$and$y = 3d^3$to obtain$(2c^2)^3 - (3d^3)^3 = (2c^2 - 3d^3)[(2c^2)^2 + (2c^2)(3d^3) + (3d^3)^2] = (2c^2 - 3d^3)(4c^4 + 6c^2 d^3 + 9d^6).\n\n#### Factoring Trinomials by Trial and Error\n\nA trinomial px^2 + qx + r$with integer coefficients$p, q, r$factors into$(ax + b)(cx + d)$, where$a, b, c, d$are integers satisfying$ac = p$,$bd = r$, and$ad + bc = q$. If no combination of integer values satisfies these conditions, the trinomial is irreducible. This method also extends to trinomials of two variables$px^2 + qxy + ry^2 = (ax + by)(cx + dy).\n\nFor example, to factor 6x^2 - 7x - 3$, find integer factors satisfying$ac = 6$,$bd = -3$, and$ad + bc = -7$. Assuming positive values for$a$and$c$, candidate pairs for$(a, c)$are$(1, 6)$,$(6, 1)$,$(2, 3)$, and$(3, 2)$. Candidate pairs for$(b, d)$satisfying$bd = -3$are$(1, -3)$,$(-1, 3)$,$(3, -1)$, and$(-3, 1)$. Testing$(2x + b)(3x + d)$with$b = -3$and$d = 1$yields$ad + bc = (2)(1) + (-3)(3) = 2 - 9 = -7$. Thus, the correct factorization is$6x^2 - 7x - 3 = (2x - 3)(3x + 1).\n\n#### Factoring by Grouping\n\nFor expressions with four or more terms, group terms into pairs or subsets to factor out common binomial expressions.\n\nTo factor 4ac + 2bc - 2ad - bd$, group the terms as$(4ac + 2bc) - (2ad + bd)$, factor out common monomials to obtain$2c(2a + b) - d(2a + b)$, and factor out the common binomial$(2a + b)$to get$(2c - d)(2a + b)$or$(2a + b)(2c - d).\n\nTo factor 3x^3 + 2x^2 - 12x - 8$, group terms as$(3x^3 + 2x^2) - (12x + 8)$, factor out common terms$x^2(3x + 2) - 4(3x + 2)$, factor out$(3x + 2)$to obtain$(x^2 - 4)(3x + 2)$, and factor$(x^2 - 4)$as a difference of squares to get$(x + 2)(x - 2)(3x + 2).\n\nTo factor x^2 - 16y^2 + 10x + 25$, rearrange and group as a perfect square trinomial minus a square term$(x^2 + 10x + 25) - 16y^2$, write as$(x + 5)^2 - (4y)^2$, apply the difference of two squares to get$[(x + 5) - 4y][(x + 5) + 4y]$, and simplify to$(x - 4y + 5)(x + 4y + 5)$$.

Bibliographic Reference

This material is referenced from College Algebra, 8th Edition, by James Stewart, Lothar Redlin, and Saleem Watson, published in 2024 (ISBN: 9780357753651).