Factoring Polynomials – Fundamental Concepts of Algebra (Precalculus Chapter P.5)

Factoring Polynomials

Introduction to Factoring

Factoring is the process of expressing a polynomial as a product of simpler polynomials, called factors. In this section, we focus on factoring over the set of integers, meaning all coefficients in the factors are integers. If a polynomial cannot be factored using integer coefficients, it is called irreducible over the integers or prime. The goal is to continue factoring until all factors are irreducible, resulting in a completely factored polynomial.

Greatest Common Factor (GCF)

Definition and Method

The greatest common factor (GCF) of a polynomial is the highest-degree expression that divides each term of the polynomial. Factoring out the GCF is typically the first step in any factoring problem, using the distributive property in reverse:

• Definition: The GCF is the largest expression (in terms of degree and numerical value) that is a factor of every term in the polynomial.

• Factoring out the GCF: $ab + ac = a(b + c)$

Example:

• Factor $2x^2(5x) - 2x^2(2)$ $= 2x^2[5x - 2]$

• Factor $(x-7)(2x+3)$ (Already factored; no further GCF extraction possible.)

Factoring by Grouping

Method

When a polynomial does not have a GCF other than 1, it may still be possible to factor by grouping terms in pairs or sets, then factoring out common factors from each group.

• Factoring by grouping: Rearrange and group terms to factor out common factors, then factor the resulting expression.

Example:

• Given $ax + ay + bx + by$, group as $(ax + ay) + (bx + by) = a(x + y) + b(x + y) = (a + b)(x + y)$

Factoring Trinomials

Trinomials of the Form $ax^2 + bx + c$

To factor trinomials, follow these steps:

1. Find two first terms whose product is $a x^2$.

2. Find two last terms whose product is $c$.

3. By trial and error, select combinations so that the sum of the outside and inside products is $bx$.

If no such combination exists, the trinomial is prime.

Example 1 (Leading Coefficient 1):

• Factor $x^2 + 13x + 40$ Factors of 40: 1, 40; 2, 20; 4, 10; 5, 8 $5 imes 8 = 40$ and $5 + 8 = 13$ So, $x^2 + 13x + 40 = (x + 5)(x + 8)$

Example 2 (Leading Coefficient 1):

• Factor $x^2 - 5x - 24$ Factors of -24: 1, -24; 2, -12; 3, -8; 4, -6 $3 imes -8 = -24$ and $3 + (-8) = -5$ So, $x^2 - 5x - 24 = (x + 3)(x - 8)$

Example 3 (Leading Coefficient not 1):

• Factor $8x^2 - 10x + 3$ Factorizations of 8: 1, 8; 2, 4 Factorizations of 3: 1, 3 Find a combination so that the sum of the inside and outside products is -10. (Process involves grouping and trial/error.)

The Difference of Two Squares

Formula and Application

The difference of two squares can always be factored as follows:

$A^2 - B^2 = (A + B)(A - B)$

• Application: The difference of the squares of two terms factors as the product of a sum and a difference of those terms.

Example:

• Factor $36x^2 - 25$ $= (6x + 5)(6x - 5)$

Factoring Perfect Square Trinomials

Formula and Recognition

A perfect square trinomial is of the form:

$A^2 + 2AB + B^2 = (A + B)^2$ $A^2 - 2AB + B^2 = (A - B)^2$

• Recognize by checking if the first and last terms are perfect squares and the middle term is twice the product of their square roots.

Example:

• Factor $16x^2 + 56x + 49$ $16x^2 = (4x)^2$, $49 = 7^2$, $56x = 2 imes 4x imes 7$ $= (4x + 7)^2$

• Factor $x^2 + 10x + 25$ $x^2 = x^2$, $25 = 5^2$, $10x = 2 imes x imes 5$ $= (x + 5)^2$

Factoring the Sum or Difference of Two Cubes

Formulas

• Sum of cubes: $A^3 + B^3 = (A + B)(A^2 - AB + B^2)$

• Difference of cubes: $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$

Example:

• Factor $x^3 + 8$ $= (x + 2)(x^2 - 2x + 4)$

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

A General Strategy for Factoring Polynomials

Step-by-Step Approach

1. If there is a common factor, factor out the GCF.

2. Determine the number of terms and try the following:

   • If two terms: check for difference of squares, sum/difference of cubes.

   • If three terms: check for perfect square trinomials, otherwise factor by trial and error.

   • If four or more terms: try factoring by grouping.

3. Check if any factors can be factored further; if so, continue until completely factored.

Example:

• Factor $3x^2 + 18x + 27$ Step 1: GCF is 3 $= 3(x^2 + 6x + 9)$ Step 2: $x^2 + 6x + 9$ is a perfect square trinomial $= 3(x + 3)^2$

Factoring Algebraic Expressions with Fractional and Negative Exponents

Method

Although not polynomials, expressions with fractional or negative exponents can be simplified using similar factoring techniques. The GCF is the term with the smallest exponent.

Example:

• Factor and simplify: $(x-1)^{1/2} + (x-1)^{-1/2}$ GCF: $(x-1)^{-1/2}$ $= (x-1)^{-1/2}[x-1 + 1] = (x-1)^{-1/2}(x)$ Or, written as a single fraction: $= \frac{2x-1}{(x-1)^{1/2}}$

Summary Table: Factoring Techniques

Type	Form	Factored Form
GCF	$ab + ac$	$a(b + c)$
Grouping	$ax + ay + bx + by$	$(a + b)(x + y)$
Trinomial (a=1)	$x^2 + bx + c$	$(x + m)(x + n)$
Trinomial (a≠1)	$ax^2 + bx + c$	$(px + q)(rx + s)$
Difference of Squares	$A^2 - B^2$	$(A + B)(A - B)$
Perfect Square Trinomial	$A^2  2AB + B^2$	$(A  B)^2$
Sum of Cubes	$A^3 + B^3$	$(A + B)(A^2 - AB + B^2)$
Difference of Cubes	$A^3 - B^3$	$(A - B)(A^2 + AB + B^2)$

Additional info: Factoring is a foundational skill in algebra, essential for solving equations, simplifying expressions, and understanding higher-level mathematics such as calculus and analytic geometry.