In these lessons, we will look at factoring by common factors and factoring of polynomials by grouping.

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The following diagram shows the steps to factor a polynomial with four terms using grouping. Scroll down the page for examples and solutions.

The following diagram shows the steps to factor a trinomial using grouping. Scroll down the page for examples and solutions.

The first step in factorizing is to find and extract the GCF of all the terms.

** Example: **

a) xyz – x

b) 6a

** Solution: **

b) 6a

Factoring is a technique that is useful when trying to solve polynomial equations algebraically.

We begin by looking for the Greatest Common Factor (GCF) of a polynomial expression.

The GCF is the largest monomial that divides (is a factor of) each term of of the polynomial.

The following video shows an example of simple factoring or factoring by common factors.

To find the GCF of a Polynomial

1. Write each term in prime factored form

2. Identify the factors common in all terms

3. Factor out the GCF

Example:

Factor out the GCF

1. 2x

2. 4x

3. -2x

4. -y

Two examples of factoring out the greatest common factor to rewrite a polynomial expression.

Examples:

Factor out the GCF:

a) 2x

b) 6a

Examples:

Factor out binomial expressions.

a) 3x

b) 5x

This video provides examples of how to factor polynomials that require factoring out the GCF as the first step. Then other methods are used to completely factor the polynomial.

Examples:

Factor

4x

3x

2x

When an expression has an even number of terms and there are no common factors for all the terms, we may group the terms into pairs and find the common factor for each pair:

** Example: **

Factorize the following expressions:

a) ax + ay + bx + byb) 2x + 8y – 3px –12py

c) 3x – 3y + 4ay – 4ax

** Solution: **

a) ax + ay + bx + by

= a(x + y) + b(x + y)

= (a + b)(x + y)

b) 2x + 8y – 3px –12py

= 2(x + 4y) –3p(x + 4y)

= (2 – 3p)(x + 4y)

c) 3x – 3y + 4ay – 4ax

= 3(x – y) + 4a(y – x)

= 3(x – y) – 4a(x – y)

= (3 – 4a)( x – y)

**How to Factor by Grouping?**

3 complete examples of solving quadratic equations using factoring by grouping are shown.

Examples:

1. Factor x(x + 1) - 5(x + 1)

2. Solve 2x^{2} + 5x + 2 = 0

3. Solve 7x^{2} + 16x + 4 = 0

4. Solve 6x^{2} - 17x + 12 = 0

Examples:

Factor:

a) 2x

b) 10x

Example:

Factor:

12u

Examples:

Factor 12x

Example:

Factor 6x

Example:

12a

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