Factoring Basics
Learning Outcomes
- Identify and factor the greatest common factor of a polynomial.
- Factor a trinomial with leading coefficient 1.
- Factor by grouping.
Factoring a GCF out of a polynomial
When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.A General Note: Greatest Common Factor
The greatest common factor (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.Distributive Property Forward and Backward
Forward: We distribute [latex]a[/latex] over [latex]b+c[/latex].
[latex]a\left(b+c\right)=ab+ac[/latex].
Backward: We factor [latex]a[/latex] out of [latex]ab+ac[/latex].[latex]ab+ac=a\left(b+c\right)[/latex].
We have seen that we can distribute a factor over a sum or difference. Now we see that we can "undo" the distributive property with factoring.Example
Factor [latex]25b^{3}+10b^{2}[/latex].Answer: Find the GCF.
[latex]\begin{array}{l}\,\,25b^{3}=5\cdot5\cdot{b}\cdot{b}\cdot{b}\\\,\,10b^{2}=5\cdot2\cdot{b}\cdot{b}\\\text{GCF}=5\cdot{b}\cdot{b}=5b^{2}\end{array}[/latex]
Rewrite each term with the GCF as one factor.[latex]\begin{array}{l}25b^{3} = 5b^{2}\cdot5b\\10b^{2}=5b^{2}\cdot2\end{array}[/latex]
Rewrite the polynomial using the factored terms in place of the original terms.[latex]5b^{2}\left(5b\right)+5b^{2}\left(2\right)[/latex]
Factor out the [latex]5b^{2}[/latex].[latex]5b^{2}\left(5b+2\right)[/latex]
Answer
[latex-display]5b^{2}\left(5b+2\right)[/latex-display]Analysis
The factored form of the polynomial [latex]25b^{3}+10b^{2}[/latex] is [latex]5b^{2}\left(5b+2\right)[/latex]. You can check this by doing the multiplication. [latex]5b^{2}\left(5b+2\right)=25b^{3}+10b^{2}[/latex]. Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over. For example:[latex]\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5.\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}.\end{array}[/latex]
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.How To: Given a polynomial expression, factor out the greatest common factor
- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Combine to find the GCF of the expression.
- Determine what the GCF needs to be multiplied by to obtain each term in the expression.
- Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
Example: Factoring the Greatest Common Factor
Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].Answer: First find the GCF of the expression. The GCF of [latex]6,45[/latex], and [latex]21[/latex] is [latex]3[/latex]. The GCF of [latex]{x}^{3},{x}^{2}[/latex], and [latex]x[/latex] is [latex]x[/latex]. (Note that the GCF of a set of expressions of the form [latex]{x}^{n}[/latex] will always be the lowest exponent.) The GCF of [latex]{y}^{3},{y}^{2}[/latex], and [latex]y[/latex] is [latex]y[/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[/latex]. Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\left(2{x}^{2}{y}^{2}\right)=6{x}^{3}{y}^{3}, 3xy\left(15xy\right)=45{x}^{2}{y}^{2}[/latex], and [latex]3xy\left(7\right)=21xy[/latex]. Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.
[latex]\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)[/latex]
Analysis of the Solution
After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].Try It
Factor [latex]x\left({b}^{2}-a\right)+6\left({b}^{2}-a\right)[/latex] by pulling out the GCF.Answer: [latex]\left({b}^{2}-a\right)\left(x+6\right)[/latex]
[ohm_question]7888[/ohm_question]Factoring a Trinomial with Leading Coefficient 1
Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial [latex]{x}^{2}+5x+6[/latex] has a GCF of 1, but it can be written as the product of the factors [latex]\left(x+2\right)[/latex] and [latex]\left(x+3\right)[/latex]. Trinomials of the form [latex]{x}^{2}+bx+c[/latex] can be factored by finding two numbers with a product of [latex]c[/latex] and a sum of [latex]b[/latex]. The trinomial [latex]{x}^{2}+10x+16[/latex], for example, can be factored using the numbers [latex]2[/latex] and [latex]8[/latex] because the product of these numbers is [latex]16[/latex] and their sum is [latex]10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].A General Note: Factoring a Trinomial with Leading Coefficient 1
A trinomial of the form [latex]{x}^{2}+bx+c[/latex] can be written in factored form as [latex]\left(x+p\right)\left(x+q\right)[/latex] where [latex]pq=c[/latex] and [latex]p+q=b[/latex].Q & A
Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime.How To: Given a trinomial in the form [latex]{x}^{2}+bx+c[/latex], factor it
- List factors of [latex]c[/latex].
- Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]c[/latex] with a sum of [latex]b[/latex].
- Write the factored expression [latex]\left(x+p\right)\left(x+q\right)[/latex].
Example: Factoring a Trinomial with Leading Coefficient 1
Factor [latex]{x}^{2}+2x - 15[/latex].Answer: We have a trinomial with leading coefficient [latex]1,b=2[/latex], and [latex]c=-15[/latex]. We need to find two numbers with a product of [latex]-15[/latex] and a sum of [latex]2[/latex]. In the table, we list factors until we find a pair with the desired sum.
| Factors of [latex]-15[/latex] | Sum of Factors |
|---|---|
| [latex]1,-15[/latex] | [latex]-14[/latex] |
| [latex]-1,15[/latex] | [latex]14[/latex] |
| [latex]3,-5[/latex] | [latex]-2[/latex] |
| [latex]-3,5[/latex] | [latex]2[/latex] |
Analysis of the Solution
We can check our work by multiplying. Use FOIL to confirm that [latex]\left(x - 3\right)\left(x+5\right)={x}^{2}+2x - 15[/latex].Q & A
Does the order of the factors matter? No. Multiplication is commutative, so the order of the factors does not matter.Try It
Factor [latex]{x}^{2}-7x+6[/latex].Answer: [latex]\left(x - 6\right)\left(x - 1\right)[/latex]
[ohm_question]7897[/ohm_question]Factoring by Grouping
Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial [latex]2{x}^{2}+5x+3[/latex] can be rewritten as [latex]\left(2x+3\right)\left(x+1\right)[/latex] using this process. We begin by rewriting the original expression as [latex]2{x}^{2}+2x+3x+3[/latex] and then factor each portion of the expression to obtain [latex]2x\left(x+1\right)+3\left(x+1\right)[/latex]. We then pull out the GCF of [latex]\left(x+1\right)[/latex] to find the factored expression.A General Note: Factoring by Grouping
To factor a trinomial of the form [latex]a{x}^{2}+bx+c[/latex] by grouping, we find two numbers with a product of [latex]ac[/latex] and a sum of [latex]b[/latex]. We use these numbers to divide the [latex]x[/latex] term into the sum of two terms and factor each portion of the expression separately then factor out the GCF of the entire expression.How To: Given a trinomial in the form [latex]a{x}^{2}+bx+c[/latex], factor by grouping
- List factors of [latex]ac[/latex].
- Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]ac[/latex] with a sum of [latex]b[/latex].
- Rewrite the original expression as [latex]a{x}^{2}+px+qx+c[/latex].
- Pull out the GCF of [latex]a{x}^{2}+px[/latex].
- Pull out the GCF of [latex]qx+c[/latex].
- Factor out the GCF of the expression.
Example: Factoring a Trinomial by Grouping
Factor [latex]5{x}^{2}+7x - 6[/latex] by grouping.Answer: We have a trinomial with [latex]a=5,b=7[/latex], and [latex]c=-6[/latex]. First, determine [latex]ac=-30[/latex]. We need to find two numbers with a product of [latex]-30[/latex] and a sum of [latex]7[/latex]. In the table, we list factors until we find a pair with the desired sum.
| Factors of [latex]-30[/latex] | Sum of Factors |
|---|---|
| [latex]1,-30[/latex] | [latex]-29[/latex] |
| [latex]-1,30[/latex] | [latex]29[/latex] |
| [latex]2,-15[/latex] | [latex]-13[/latex] |
| [latex]-2,15[/latex] | [latex]13[/latex] |
| [latex]3,-10[/latex] | [latex]-7[/latex] |
| [latex]-3,10[/latex] | [latex]7[/latex] |
[latex]\begin{array}{cc}5{x}^{2}-3x+10x - 6 \hfill & \text{Rewrite the original expression as }a{x}^{2}+px+qx+c.\hfill \\ x\left(5x - 3\right)+2\left(5x - 3\right)\hfill & \text{Factor out the GCF of each part}.\hfill \\ \left(5x - 3\right)\left(x+2\right)\hfill & \text{Factor out the GCF}\text{ }\text{ of the expression}.\hfill \end{array}[/latex]
Analysis of the Solution
We can check our work by multiplying. Use FOIL to confirm that [latex]\left(5x - 3\right)\left(x+2\right)=5{x}^{2}+7x - 6[/latex].Try It
Factor the following.- [latex]2{x}^{2}+9x+9[/latex]
- [latex]6{x}^{2}+x - 1[/latex]
Answer:
- [latex]\left(2x+3\right)\left(x+3\right)[/latex]
- [latex]\left(3x - 1\right)\left(2x+1\right)[/latex]
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Example: Greatest Common Factor. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Factoring Trinomials by Grouping. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID 7886, 7897, 7908. Authored by: Tyler Wallace. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
CC licensed content, Specific attribution
- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
