Factoring Polynomials: Methods and Applications

Factoring Polynomials

Introduction

Factoring is a fundamental algebraic skill that involves expressing a polynomial as a product of simpler polynomials. Mastery of factoring techniques is essential for solving equations, simplifying expressions, and understanding polynomial functions.

Common Monomial Factors

Identifying and Factoring Out Common Monomial Factors

• Definition: A monomial factor is a single term (number, variable, or product of both) that divides each term of a polynomial.

• Factoring Process: Identify the greatest common factor (GCF) among all terms and factor it out.

Example Table:

Polynomial	Common Monomial Factor	Remaining Factor	Factored Form
4x + 2	2	2x + 1	2(2x + 1)
5x – 10	5	x – 2	5(x – 2)
8x2 + 4x + 2	2	4x2 + 2x + 1	2(4x2 + 2x + 1)
12x – 20	4	3x – 5	4(3x – 5)
x3 + x	x	x2 + 1	x(x2 + 1)
5x3 – 4x2	x2	5x – 4	x2(5x – 4)
8x2 + 12x	4x	2x + 3	4x(2x + 3)

Special Factoring Patterns

Difference of Two Squares

• Formula: $a^2 - b^2 = (a - b)(a + b)$

• Example: $x^2 - 25 = (x - 5)(x + 5)$

Difference of Two Cubes

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

• Example: $x^3 - 27 = (x - 3)(x^2 + 3x + 9)$

Sum of Two Cubes

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

• Example: $x^3 + 64 = (x + 4)(x^2 - 4x + 16)$

Factoring Higher Degree Differences of Squares

• Apply the difference of squares repeatedly if possible.

• Example: $x^4 - 625 = (x^2 - 25)(x^2 + 25) = (x - 5)(x + 5)(x^2 + 25)$

Perfect Square Trinomials

• Formula: $a^2 + 2ab + b^2 = (a + b)^2$

• Formula: $a^2 - 2ab + b^2 = (a - b)^2$

• Examples:

  • $x^2 + 8x + 16 = (x + 4)^2$

  • $4x^2 + 12x + 9 = (2x + 3)^2$

  • $9x^2 + 24x + 16 = (3x + 4)^2$

Factoring Trinomials

Factoring $x^2 + Bx + C$ (A = 1)

• Find integers $a$ and $b$ such that $ab = C$ and $a + b = B$.

• Formula: $x^2 + Bx + C = (x + a)(x + b)$

• Examples:

  • $x^2 + 9x + 14 = (x + 2)(x + 7)$

  • $x^2 - 7x + 10 = (x - 2)(x - 5)$

  • $x^2 - x - 20 = (x + 4)(x - 5)$

  • $x^2 + 2x - 15 = (x - 3)(x + 5)$

Prime Polynomials

• A polynomial is prime if it cannot be factored over the real numbers.

• Theorem: Any polynomial of the form $x^2 + a^2$ (where $a$ is real) is prime.

• Example: $x^2 + 8$ is prime because no integer pairs multiply to 8 and sum to 0.

Factoring by Grouping

Factoring by Grouping

• Group terms to factor out common binomial or monomial factors.

• Examples:

  • $(2x + 1)x + (2x + 1)\cdot5 = (2x + 1)(x + 5)$

  • $5(x - 2)^3(x + 1)^3 + 4(x - 2)^4(x + 1)^2 = (x - 2)^3(x + 1)^2(9x - 3)$

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

Factoring $Ax^2 + Bx + C$ (A ≠ 1)

Steps for Factoring When $A \neq 1$

1. Find $AC$.

2. Find integers $a$ and $b$ such that $ab = AC$ and $a + b = B$.

3. Rewrite $Bx$ as $ax + bx$.

4. Factor by grouping.

Example 1: $3x^2 + 10x + 3$

• $AC = 9$; $a = 1$, $b = 9$

• $3x^2 + 9x + x + 3 = (3x^2 + 9x) + (x + 3) = 3x(x + 3) + 1(x + 3) = (x + 3)(3x + 1)$

Example 2: $3x^2 - x - 4$

• $AC = -12$; $a = 3$, $b = -4$

• $3x^2 + 3x - 4x - 4 = (3x^2 + 3x) + (-4x - 4) = 3x(x + 1) - 4(x + 1) = (x + 1)(3x - 4)$

Completing the Square

Completing the Square for $x^2 + bx$

• Identify the coefficient $b$ of $x$.

• Compute $\left(\frac{b}{2}\right)^2$ and add it to $x^2 + bx$.

• The expression becomes a perfect square trinomial: $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

Examples:

Start	Add	Result	Factored Form
$y^2 + 6y$	$9$	$y^2 + 6y + 9$	$(y + 3)^2$
$x^2 + 10x$	$25$	$x^2 + 10x + 25$	$(x + 5)^2$
$a^2 - 4a$	$4$	$a^2 - 4a + 4$	$(a - 2)^2$
$p^2 - 7p$	$\frac{49}{4}$	$p^2 - 7p + \frac{49}{4}$	$(p - \frac{7}{2})^2$