Techniques for Factoring Polynomials

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More Lessons for Algebra

Math Worksheets

A series of free Basic Algebra Lessons.

In this lesson, we will learn

• how to factor a polynomial by factoring out the greatest common factor.
• how to factor a polynomial by grouping.
• how to factor difference of perfect squares.
• how to factor trinomials with a = 1
• how to factor trinomials with a > 1

Greatest Common Factors

When working with polynomials and complex fractions, it's important to understand and be able to find greatest common factors. Being able to find greatest common factors will help when factoring trinomials by grouping. When factoring, we can either find the greatest common factors of two integers or find the greatest common factors of two complex expressions.
Factoring out the greatest common factor.
Factor by Grouping

Difference of Perfect Squares

An important special case when trying to factor polynomials is a identifying the difference of perfect squares. We learn to recognize a difference of perfect squares because they have a special, easily factored form. It's also important to recognize the factored form to make the multiplication of the binomials easier. Easily recognizing the difference of perfect squares is useful when factoring quadratics that are not a difference of perfect squares.
Factoring the Difference of Two Squares - Ex 1
Factoring the Difference of Two Squares - Ex 2

Factoring the Difference of Two Squares - Ex 3

Factoring Trinomials, a = 1

When given a trinomial, or a quadratic, it can be useful for purposes of canceling and simplifying to factor it. Factoring trinomials is easiest when the leading coefficient (the coefficient on the squared term) is one. A more complex situation is factoring trinomials when the leading coefficient is not one. To factor trinomials we use methods that involve finding the factors of their coefficients.
Factoring a Trinomial with Leading Coefficient of 1 - The Basics

Factoring Basic Trinomials, with a = 1

Factoring Trinomials with "a" Not Equal to One

One of the final steps in learning to factor trinomials is factoring trinomials with "a" not equal to one. In this case, "a" is the leading coefficient, or the coefficient of the squared term. When factoring trinomials with "a" not equal to one, in addition to using the methods used when "a" is one we must take the factors of "a" into account when finding the terms of the factored binomials.
This video provides examples of how to factor a trinomial when the leading coefficient is not equal to 1 by using the grouping method

This video provides examples of how to factor a trinomial when the leading coefficient is not equal to 1 by using the trial and error method.
This video provides examples of how to factor a trinomial when the leading coefficient is not equal to 1 by using the bottoms up method.

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