Factoring by Common Factors & by Grouping

Factor Trinomials by Grouping
Factoring trinomials by grouping (often called the “ac method”) is a technique that is especially useful when the leading coefficient (the ‘a’ in ax² + bx + c) is not 1. It transforms a trinomial into a four-term polynomial, which can then be factored by grouping.

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 how to factor a trinomial by grouping. Scroll down the page for examples and solutions.

 

Algebra Worksheets
Practice your skills with the following Algebra worksheets:
Printable & Online Algebra Worksheets

Steps to Factor Trinomials by Grouping

1. Check for a Greatest Common Factor (GCF):
   Look for a GCF that divides all three terms (ax², bx, and c). If there is one, factor it out.
2. Find the “ac” Product:
   Multiply the coefficient of the x² term (a) by the constant term (c).
3. Find Two Numbers that Multiply to “ac” and Add to “b”:
   List out all the factor pairs of your ac product.
   From that list, identify the pair of numbers that also adds up to the coefficient of the middle term (b). Be careful of the signs.
4. Rewrite the Middle Term (bx) using the Two Numbers:
   Replace the original middle term (bx) with two new terms, using the two numbers you found in step 3 as their coefficients.
5. Group the Four Terms into Two Pairs:
   Put parentheses around the first two terms and around the last two terms. If the third term has a negative sign, be very careful to include it with the grouping.
6. Factor Out the GCF from Each Pair:
   Find the GCF of the first pair and factor it out.
   Find the GCF of the second pair and factor it out.
   If you’ve done everything correctly, the binomial remaining in the parentheses after factoring out the GCF should be the same for both pairs. If they are not identical, go back and check your work.
7. Factor Out the Common Binomial:
   Since the binomial in parentheses is the same for both terms, factor it out as a common factor. The terms left over (the GCFs you factored out in step 6) will form the second binomial.
8. Check Your Answer:
   Multiply your two binomial factors back together (using FOIL or distributive property) to ensure you get the original trinomial.

Factoring By Common Factors

The first step in factorizing is to find and extract the GCF of all the terms.
Example:
Factorize the following algebraic expressions:
a) xyz – x²z
b) 6a²b + 4bc

Solution:
a) xyz – x²z = xz(y – x)
b) 6a²b + 4bc = 2b(3a² + 2c)

Factoring Out The Greatest Common Factor
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

Examples:
Factor out the GCF

1. 2x⁴ - 16x³
2. 4x²y³ + 20xy² + 12xy
3. -2x³ + 8x² - 4x
4. -y³ - 2y² + y - 7

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Factoring Using the Great Common Factor, GCF - Example 1
Two examples of factoring out the greatest common factor to rewrite a polynomial expression.

Example:
Factor out the GCF:
a) 2x³y⁸ + 6x⁴y² + 10x⁵y¹⁰
b) 6a¹⁰b⁸ + 3a⁷b⁴ - 24a⁵b⁶

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Factoring Using the Great Common Factor, GCF - Example 2
Example:
Factor out binomial expressions.
a) 3x²(2x + 5y) + 7y²(2x + 5y)
b) 5x²(x + 3y) - 15x³(x + 3y)

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Factoring Polynomials with Common Factors
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.

Example:
Factor
4x² - 64
3x² + 3x - 36
2x² - 28x + 98

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Factoring By Grouping

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 + by
b) 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 2x² + 5x + 2 = 0
   
3. Solve 7x² + 16x + 4 = 0
   
4. Solve 6x² - 17x + 12 = 0

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Factoring by Grouping - Ex 1
Example:
Factor:
a) 2x² + 7x² + 2x + 7
b) 10x² + 2xy + 15xy + 3y²

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Factoring By Grouping - Ex 2
Example:
Factor:
12u² + 15uv + 24uv² + 30v³

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Factoring Trinomials: Factor by Grouping - ex 1
Example:
Factor 12x² + 34x + 10

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Factoring Trinomials: Factor by Grouping - ex 2
Example:
Factor 6x² + 15x - 21

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Factoring by grouping - Prime Factorization

Example:
12a³ - 9a²b - 8ab² + 6b³

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