Lesson 6.4: General Strategy for Factoring Polynomials

New Concept

This lesson ties everything together! You'll learn a master strategy to decide which factoring method to use for any polynomial, ensuring you can break it down completely and confidently, from finding the GCF to applying special patterns.

What’s next

Next, you'll see this strategy in action through worked examples and then test your skills on a series of challenge problems.

Property

Step 1. Is there a greatest common factor? Factor it out.

Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?

• If it is a binomial: Is it a sum of squares (prime), sum of cubes, difference of squares, or difference of cubes?
• If it is a trinomial: Is it of the form $x^2 + bx + c$ or $ax^2 + bx + c$? Check for perfect square patterns.
• If it has more than three terms: Use the grouping method.

Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial? A polynomial is completely factored if, other than monomials, its factors are prime.

Property

• Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$
• Sum of Squares: Sums of squares do not factor.
• Sum of Cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
• Difference of Cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Examples

• To factor the difference of squares $9x^2 - 49$, recognize it as $(3x)^2 - 7^2$. Applying the formula gives $(3x-7)(3x+7)$.

• To factor the sum of cubes $8a^3 + 27b^3$, view it as $(2a)^3 + (3b)^3$. The formula yields $(2a+3b)(4a^2 - 6ab + 9b^2)$.

Property

• Trinomials of the form $x^2 + bx + c$: Find two integers that multiply to $c$ and add to $b$, then 'Undo FOIL'.
• Trinomials of the form $ax^2 + bx + c$: Use the trial and error or 'ac' method. If $a$ and $c$ are perfect squares, check if it fits a perfect square trinomial pattern.
• Perfect Square Trinomials:

Examples

• To factor $x^2 - 2x - 15$, find two numbers that multiply to $-15$ and add to $-2$. These are $-5$ and $3$, so the factors are $(x-5)(x+3)$.

• To factor the perfect square trinomial $9x^2 + 12x + 4$, notice it fits the pattern $(3x)^2 + 2(3x)(2) + 2^2$. This factors directly to $(3x+2)^2$.

Property

If a polynomial has more than three terms, use the grouping method.

Step 1. Group the polynomial into pairs of terms.

Step 2. Factor out the GCF from each pair.

Property

A polynomial is completely factored if, other than monomials, its factors are prime. To factor completely:

Step 1. Always factor out the Greatest Common Factor (GCF) first.

Step 2. After factoring, examine each new factor to see if it can be factored further. For example, a difference of squares might appear after the initial step.

Property

When a four-term polynomial cannot be factored by pairing, try grouping three terms that form a perfect square trinomial.

Step 1. Identify and factor the perfect square trinomial, resulting in the form $(a ± b)^2$.

Step 2. The expression becomes a difference of two squares: $(a ± b)^2 - c^2$.