Algebra: Factoring
Contents (8 topics • 43 videos)
Factoring an Algebraic Expression Using the Greatest Common Factor (GCF)
Introduction Videos
Determining the Greatest Common Factor of Algebra Terms 
Factoring Using the Greatest Common Factor
Factor using the GCF.
\[ 14y^3-21y^2+7y \]
Factor using the GCF.
\[ -mn^2-mn+mn^3 \]
Factor using the GCF.
\[ -48g^8 h^3-24g^6 h^4+60g^4 h^5-12g^2 h^6 \]
Multiply, combine like terms, and then factor the expression using the GCF.
\[ 10x^2(3y-y^3)-4y(8x^3-5x)+32x^3y \]
Documents
Practice Problems (.PDF) 
Practice Problems - Solutions (.PDF) 
Factoring by Grouping
Introduction Videos
Factoring by Grouping
Factor by grouping.
\[ 9y(y-1)+4(y-1) \]
Factor by grouping.
\[ 49c^3+35c^2+28c+20 \]
Factor by grouping.
\[ 32a^3b-24a^3-112b^3a^2+84b^2a^2 \]
Factor by grouping.
\[ 45a^2 b+63a^2 m-120amb-168am^2 \]
Factor by grouping.
\[ x^5-2x^4+4x^3+5x^2-10x+20 \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Factoring Quadratics Expressions of the Form \( x^2+bx+c \)
Introduction Videos
Factoring Quadratic Expressions of the Form \( x^2+bx+c \)
Factor.
\[ b^2+9b+8 \]
Factor.
\[ k^2-14k+49 \]
Factor completely.
\[ -y^3+13y^2-30y \]
Factor completely.
\[ x^3+4x^2+4x+2x^2+8x+8 \]
Factor completely.
\[ 5x^2(x^2+6x+8)-10x(x^2+6x+8)-120(x^2+6x+8) \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Factoring Quadratic Expressions: The "AC" Method
Introduction Videos
The "AC" Method
Factor.
\[ 7b^2+17b+6 \]
Factor.
\[ 6g^2-13g-5 \]
Factor.
\[ 10j^2-7jk+k^2 \]
Factor.
\[ 35u^2-2u-1 \]
Factor completely.
\[ -15x^3-85x^2y+450xy^2 \]
Factor completely.
\[ 5a(6a^2+19a+10)+3(6a^2+19a+10) \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Difference of Squares Factorization
Introduction Videos
Difference of Squares Factorization
Theorems and Definitions
Difference of Squares Factorization
The difference of squares factorization is given by
\( a^2-b^2=(a-b)(a+b) \)
Factor.
\[ 10h^7-360h^5 \]
Factor.
\[ m^3+3m^2-9m-27 \]
Factor.
\[ x^3-125x \]
Factor completely.
\[ 5x^5-25x^4-125x^3y^4+20x^3+625x^2y^4-500xy^4 \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Sum and Difference of Cubes Factorizations
Introduction Videos
Sum and Difference of Cubes Factorization
Theorems and Definitions
Difference of Cubes Factorization
The difference of cubes factorization is given by
\( a^3-b^3=(a-b)(a^2+ab+b^2) \)
Sum of Cubes Factorization
The sum of cubes factorization is given by
\( a^3+b^3=(a+b)(a^2-ab+b^2) \)
Factor.
\[ a^3+27 \]
Factor.
\[ 125f^6-1 \]
Factor.
\[ 27m^{12}n^6+125m^3 \]
Factor completely.
\[ 64p^5-112p^4-120p^3-8p^2+14p+15 \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Choosing a Factoring Method
Introduction Videos
Choosing a Factoring Method
Theorems and Definitions
Choosing a Factoring Method
The general steps to factoring an algebraic expression include:
Start by factoring any greatest common factor (GCF) from the expression.
If there are two terms in the expression, see if the factorization is a difference of squares, difference of cubes, or sum of cubes. If it can be factored as both the difference of squares and sum/difference of cubes, start with the difference of squares first.
If there are three terms, see if the expression can be factored as a quadratic trinomial or using the "AC" Method.
If there are four or more terms, see if the expression can be factored using grouping.
After an initial factorization, it is possible that the resulting factors can be factored again by using one of the above methods.
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Solving Equations by Factoring
Introduction Videos
Solving Equations by Factoring
Theorems and Definitions
Zero Property of Multiplication
For an equation of the form \( pq=0 \), either \( p=0 \) or \( q=0 \).
This property serves as the foundation for solving equations by factoring. In order for the property to work, the equation must be set equal to 0.
One additional note, if the factored expression has a greatest common factor, then one of two things must be true:
- If the greatest common factor contains a variable, then the solution set will include 0.
- If the greatest common factor does not contain a variable, then the solution set will not include 0.
Solve by factoring.
\[ a^3-4a^2-21a=0 \]
Solve by factoring.
\[ 2c^5-32c=0 \]
Solve by factoring.
\[ x^5+x^3 (5x+6)-4x^3=4x(5x+6) \]
Solve by factoring.
\[ t^2+36=12t \]
Solve by factoring.
\[ 49y^4=25y^2 \]
Solve by factoring.
\[ -8q^3-38q^2=-10q \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
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