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Factoring by Common Factors & by Grouping




 

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

Related Topics:
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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(yx)
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
Example:
Factor out the GCF
1. 2x4 - 16x3
2. 4x2y3 + 20xy2 + 12xy
3. -2x3 + 8x2 - 4x
4. -y3 - 2y2 + y - 7



Factoring Using the Great Common Factor, GCF - Example 1
Two examples of factoring out the greatest common factor to rewrite a polynomial expression.
Examples:
Factor out the GCF:
a) 2x3y8 + 6x4y2 + 10x5y10
b) 6a10b8 + 3a7b4 - 24a5b6
Factoring Using the Great Common Factor, GCF - Example 2
Examples:
Factor out binomial expressions.
a) 3x2(2x + 5y) + 7y2(2x + 5y)
b) 5x2(x + 3y) - 15x3(x + 3y)
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.
Examples:
Factor
4x2 - 64
3x2 + 3x - 36
2x2 - 28x + 98

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(xy) + 4a(yx)
= 3(xy) – 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
Factoring by Grouping - Ex 1
Examples:
Factor:
a) 2x2 + 7x2 + 2x + 7
b) 10x2 + 2xy + 15xy + 3y2
Factoring By Grouping - Ex 2
Example:
Factor:
12u2 + 15uv + 24uv2 + 30v3
Factoring Trinomials: Factor by Grouping - ex 1
Examples:
Factor 12x2 + 34x + 10


Factoring Trinomials: Factor by Grouping - ex 2
Example:
Factor 6x2 + 15x - 21
Factoring by grouping - Prime Factorization
Example:
12a3 - 9a2b - 8ab2 + 6b3

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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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