Factoring Polynomials: A Study Guide for College Algebra

Section R.6: Factoring Polynomials

Factoring polynomials is a fundamental skill in College Algebra, allowing us to rewrite expressions as products of simpler factors. This process is essential for solving equations, simplifying expressions, and understanding polynomial functions.

Factoring Out the Greatest Common Factor (GCF)

Factoring out the greatest common factor (GCF) is often the first step in factoring any polynomial. The GCF is the largest expression that divides each term of the polynomial.

• Definition: The GCF of a set of terms is the largest expression (number, variable, or combination) that divides each term exactly.

• Process:

  1. Identify the GCF of all terms in the polynomial.

  2. Factor the GCF out by dividing each term by the GCF and writing the result in parentheses.

  3. If the leading term is negative, factor out a negative GCF for standard form.

• Example: Factor $6x^2 + 8x$ GCF is $2x$: $6x^2 + 8x = 2x(3x + 4)$

Factoring Trinomials

Trinomials are polynomials with three terms, often in the form $ax^2 + bx + c$. Factoring trinomials involves finding two binomials whose product is the original trinomial.

• FOIL Process: Factoring trinomials is the reverse of the FOIL (First, Outer, Inner, Last) multiplication process.

• Standard Form: $x^2 + bx + c$ factors as $(x + m)(x + n)$, where $m$ and $n$ are numbers that multiply to $c$ and add to $b$.

• Example: Factor $x^2 + 5x + 6$ Find $m$ and $n$ such that $m imes n = 6$ and $m + n = 5$. $x^2 + 5x + 6 = (x + 2)(x + 3)$

• If the leading coefficient $a \neq 1$, use factoring by grouping or the "ac method".

Factoring a Difference of Squares

A difference of squares is an expression of the form $a^2 - b^2$. It can always be factored into the product of two conjugate binomials.

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

• Example: $9x^2 - 16 = (3x + 4)(3x - 4)$

• This method only works for subtraction (difference), not addition (sum).

Factoring a Difference or Sum of Cubes

Expressions involving cubes can be factored using specific formulas for the difference or sum of cubes.

• Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

• Sum of Cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

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

• Use the acronym SOAP to remember the signs:

  • Same sign as in the original problem

  • Opposite sign

  • Always Positive (the last term)

Factoring Strategy

To factor any polynomial completely, follow this systematic approach:

1. Check for a GCF and factor it out first.

2. If the remaining expression has three terms, try to reverse the FOIL process (factor as a trinomial).

3. If the remaining expression has two terms, check if it is a difference of squares, a sum of cubes, or a difference of cubes, and use the appropriate formula.

Summary Table: Factoring Formulas

Type	General Form	Factored Form
GCF	$a x^n + a y^n$	$a(x^n + y^n)$
Trinomial (simple)	$x^2 + bx + c$	$(x + m)(x + n)$
Difference of Squares	$a^2 - b^2$	$(a + b)(a - b)$
Sum of Cubes	$a^3 + b^3$	$(a + b)(a^2 - ab + b^2)$
Difference of Cubes	$a^3 - b^3$	$(a - b)(a^2 + ab + b^2)$

Practice Problems (Selected)

• Factor out the GCF: $8x^4 + 24x$

• Factor the trinomial: $x^2 + 9x + 20$

• Factor the difference of squares: $9x^2 - 16$

• Factor the sum of cubes: $x^3 + 27$

Additional info: Factoring is a foundational algebraic skill that is used in solving equations, simplifying rational expressions, and analyzing polynomial functions. Mastery of these techniques is essential for success in higher-level mathematics.