Lesson 5: General Strategy for Factoring Polynomials

New Concept

This lesson unifies your factoring skills into a single strategy. You'll learn to analyze any polynomial by its number of terms, then choose the correct method—from GCF to grouping—to factor it completely.

What’s next

Now that you have the roadmap, you'll apply it. Next up are interactive examples and a series of practice cards to sharpen your factoring decisions.

Property

HOW TO: Factor polynomials.

1. Step 1. Is there a greatest common factor? Factor it out.
2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms? If it is a binomial, check for a sum/difference of squares or cubes. If it is a trinomial, check if it is of the form $x^2 + bx + c$ or $ax^2 + bx + c$. If it has more than three terms, use the grouping method.
3. Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

Examples

• To factor $5x^6 + 15x^5$, first find the GCF, which is $5x^5$. Factoring it out gives $5x^5(x+3)$. The binomial $(x+3)$ is prime, so the factoring is complete.

• To factor $4x^2 - 8x - 60$, first factor out the GCF of 4, which gives $4(x^2 - 2x - 15)$. Then factor the trinomial to get $4(x-5)(x+3)$.

Property

Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$

Sum of Squares: Sums of squares do not factor.

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

Property

Trinomial of the form $x^2 + bx + c$: Undo FOIL by finding two numbers that multiply to $c$ and add to $b$. The form is $(x  )(x  )$.

Trinomial of the form $ax^2 + bx + c$:

• If $a$ and $c$ are squares, check for a Perfect Square Trinomial pattern: $(a+b)^2 = a^2 + 2ab + b^2$ or $(a-b)^2 = a^2 - 2ab + b^2$.
• Otherwise, use the trial and error or 'ac' method.

Examples

• The trinomial $9x^2 - 30xy + 25y^2$ is a perfect square trinomial because $(3x)^2 - 2(3x)(5y) + (5y)^2$. It factors into $(3x-5y)^2$.

Property

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

How to check:

1. After an initial factoring step (like GCF or difference of squares), examine each new factor.
2. Can any of the new factors be factored again?
3. Continue factoring until all factors are prime.

Examples

• To factor $5y^4 - 80$, first take out the GCF of 5: $5(y^4 - 16)$. Then factor the difference of squares: $5(y^2-4)(y^2+4)$. Factor again: $5(y-2)(y+2)(y^2+4)$.