Finding the Greatest Common Factor of a Polynomial
Learning Outcomes
- Factor the greatest common monomial out of a polynomial
Polynomial | Terms | Common Factors |
---|---|---|
[latex]6x+9[/latex] | [latex]6x[/latex] and [latex]9[/latex] | [latex]3[/latex] is a factor of [latex]6x[/latex] and [latex]9[/latex] |
[latex]a^{2}–2a[/latex] | [latex]a^{2}[/latex] and [latex]−2a[/latex] | a is a factor of [latex]a^{2}[/latex] and [latex]−2a[/latex] |
[latex]4c^{3}+4c[/latex] | [latex]4c^{3}[/latex] and [latex]4c[/latex] | [latex]4[/latex] and c are factors of [latex]4c^{3}[/latex] and [latex]4c[/latex] |
[latex]\begin{array}{ccc}\hfill 2\left(x + 7\right)&\text{factors}\hfill \\ \hfill 2\cdot x + 2\cdot 7\hfill \\ \hfill 2x + 14&\text{product}\hfill \end{array}[/latex]
Here, we will start with a product, like [latex]2x+14[/latex], and end with its factors, [latex]2\left(x+7\right)[/latex]. To do this we apply the Distributive Property "in reverse". To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in a factored form. Recall that the distributive property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.Distributive Property
If [latex]a,b,c[/latex] are real numbers, then [latex-display]a\left(b+c\right)=ab+ac\text{ and }ab+ac=a\left(b+c\right)[/latex-display]Distributive Property Forward and Backward
Forward: Product of a number and a sum: [latex]a\left(b+c\right)=a\cdot{b}+a\cdot{c}[/latex]. You can say that “[latex]a[/latex] is being distributed over [latex]b+c[/latex].” Backward: Sum of the products: [latex]a\cdot{b}+a\cdot{c}=a\left(b+c\right)[/latex]. Here you can say that “a is being factored out.”example
Factor: [latex]2x+14[/latex] SolutionStep 1: Find the GCF of all the terms of the polynomial. | Find the GCF of [latex]2x[/latex] and [latex]14[/latex]. | |
Step 2: Rewrite each term as a product using the GCF. | Rewrite [latex]2x[/latex] and [latex]14[/latex] as products of their GCF, [latex]2[/latex]. [latex-display]2x=2\cdot x[/latex-display] [latex]14=2\cdot 7[/latex] | [latex]2x+14[/latex] [latex]\color{red}{2}\cdot x+\color{red}{2}\cdot7[/latex] |
Step 3: Use the Distributive Property 'in reverse' to factor the expression. | [latex]2\left(x+7\right)[/latex] | |
Step 4: Check by multiplying the factors. | Check: [latex-display]2(x+7)[/latex-display] [latex-display]2\cdot{x}+2\cdot{7}[/latex-display] [latex]2x+14\quad\checkmark[/latex] |
try it
[ohm_question]146330[/ohm_question][latex]\begin{array}{cccc}\text{Noun}\hfill & & & 7\text{ is a factor of }14\hfill \\ \text{Verb}\hfill & & & \text{factor }2\text{ from }2x+14\hfill \end{array}[/latex]
Factor the greatest common factor from a polynomial
- Find the GCF of all the terms of the polynomial.
- Rewrite each term as a product using the GCF.
- Use the Distributive Property ‘in reverse’ to factor the expression.
- Check by multiplying the factors.
example
Factor: [latex]3a+3[/latex]Answer: Solution
[latex]3a+3[/latex] | |
Rewrite each term as a product using the GCF. | [latex]\color{red}{3}\cdot a+\color{red}{3}\cdot1[/latex] |
Use the Distributive Property 'in reverse' to factor the GCF. | [latex]3(a+1)[/latex] |
Check by multiplying the factors to get the original polynomial. | |
[latex]3(a+1)[/latex] [latex-display]3\cdot{a}+3\cdot{1}[/latex-display] [latex]3a+3\quad\checkmark[/latex] |
try it
[ohm_question]146331[/ohm_question]example
Factor: [latex]12x - 60[/latex]Answer: Solution
[latex]12x-60[/latex] | |
Rewrite each term as a product using the GCF. | [latex]\color{red}{12}\cdot x-\color{red}{12}\cdot 5[/latex] |
Factor the GCF. | [latex]12(x-5)[/latex] |
Check by multiplying the factors. | |
[latex]12(x-5)[/latex] [latex-display]12\cdot{x}-12\cdot{5}[/latex-display] [latex]12x-60\quad\checkmark[/latex] |
try it
[ohm_question]146332[/ohm_question]example
Factor: [latex]3{y}^{2}+6y+9[/latex]Answer: Solution
[latex]3y^2+6y+9[/latex] | |
Rewrite each term as a product using the GCF. | [latex]\color{red}{3}\cdot{y}^{2}+\color{red}{3}\cdot 2y+\color{red}{3}\cdot 3[/latex] |
Factor the GCF. | [latex]3(y^{2}+2y+3)[/latex] |
Check by multiplying. | |
[latex]3(y^{2}+2y+3)[/latex] [latex-display]3\cdot{y^2}+3\cdot{2y}+3\cdot{3}[/latex-display] [latex]3y^{2}+6y+9\quad\checkmark[/latex] |
try it
[ohm_question]146333[/ohm_question]example
Factor: [latex]6{x}^{2}+5x[/latex]Answer: Solution
[latex]6{x}^{2}+5x[/latex] | |
Find the GCF of [latex]6{x}^{2}[/latex] and [latex]5x[/latex] and the math that goes with it. | |
Rewrite each term as a product. | [latex]\color{red}{x}\cdot{6x}+\color{red}{x}\cdot{5}[/latex] |
Factor the GCF. | [latex]x\left(6x+5\right)[/latex] |
Check by multiplying. | |
[latex]x\left(6x+5\right)[/latex] [latex-display]x\cdot 6x+x\cdot 5[/latex-display] [latex]6{x}^{2}+5x\quad\checkmark[/latex] |
try it
[ohm_question]146335[/ohm_question]example
Factor: [latex]4{x}^{3}-20{x}^{2}[/latex]Answer: Solution
[latex]4x^3-20x^2[/latex] | ||
Rewrite each term. | [latex]\color{red}{4{x}^{2}}\cdot x - \color{red}{4{x}^{2}}\cdot 5[/latex] | |
Factor the GCF. | [latex]4x^2(x-5)[/latex] | |
Check. | [latex]4x^2(x-5)[/latex] [latex-display]4x^2\cdot{x}-4x^2\cdot{5}[/latex-display] [latex-display]4x^3-20x^2\quad\checkmark[/latex-display] |
Example
Factor [latex]25b^{3}+10b^{2}[/latex].Answer: Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex] to be [latex]5b^{2}[/latex].
[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][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.try it
[ohm_question]146337[/ohm_question]example
Factor: [latex]21{y}^{2}+35y[/latex]Answer: Solution
Find the GCF of [latex]21{y}^{2}[/latex] and [latex]35y[/latex] | ||
[latex]21y^2+35y[/latex] | ||
Rewrite each term. | [latex]\color{red}{7y}\cdot 3y + \color{red}{7y}\cdot 5[/latex] | |
Factor the GCF. | [latex]7y(3y+5)[/latex] |
try it
[ohm_question]146338[/ohm_question]example
Factor: [latex]14{x}^{3}+8{x}^{2}-10x[/latex]Answer: Solution Previously, we found the GCF of [latex]14{x}^{3},8{x}^{2},\text{and}10x[/latex] to be [latex]2x[/latex].
[latex]14{x}^{3}+8{x}^{2}-10x[/latex] | |
Rewrite each term using the GCF, 2x. | [latex]\color{red}{2x}\cdot 7{x}^{2}+\color{red}{2x}\cdot4x-\color{red}{2x}\cdot 5[/latex] |
Factor the GCF. | [latex]2x\left(7{x}^{2}+4x - 5\right)[/latex] |
[latex]2x(7x^2+4x-5)[/latex] [latex-display]2x\cdot{7x^2}+2x\cdot{4x}-2x\cdot{5}[/latex-display] [latex]14x^3+8x^2-10x\quad\checkmark[/latex] |
try it
[ohm_question]146339[/ohm_question]example
Factor: [latex]-9y - 27[/latex]Answer: Solution
When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of [latex]9y[/latex] and [latex]27[/latex] is [latex]9[/latex]. | |
Since the expression [latex]−9y−27[/latex] has a negative leading coefficient, we use [latex]−9[/latex] as the GCF. | |
[latex]-9y - 27[/latex] | |
Rewrite each term using the GCF. | [latex]\color{red}{-9}\cdot y + \color{red}{(-9)}\cdot 3[/latex] |
Factor the GCF. | [latex]-9\left(y+3\right)[/latex] |
Check. [latex-display]-9(y+3)[/latex-display] [latex-display]-9\cdot{y}+(-9)\cdot{3}[/latex-display] [latex]-9y-27\quad\checkmark[/latex] |
try it
[ohm_question]146340[/ohm_question]example
Factor: [latex]-4{a}^{2}+16a[/latex]Answer: Solution
The leading coefficient is negative, so the GCF will be negative. | |
Since the leading coefficient is negative, the GCF is negative, [latex]−4a[/latex]. | |
[latex]-4{a}^{2}+16a[/latex] | |
Rewrite each term. | [latex]\color{red}{-4a}\cdot{a}-\color{red}{(-4a)}\cdot{4}[/latex] |
Factor the GCF. | [latex]-4a\left(a - 4\right)[/latex] |
Check on your own by multiplying. |
TRY IT
[ohm_question]146341[/ohm_question]Example
Factor [latex]81c^{3}d+45c^{2}d^{2}[/latex].Answer: Factor [latex]81c^{3}d[/latex].
[latex]3\cdot3\cdot9\cdot{c}\cdot{c}\cdot{c}\cdot{d}[/latex]
Factor [latex]45c^{2}d^{2}[/latex].[latex]3\cdot3\cdot5\cdot{c}\cdot{c}\cdot{d}\cdot{d}[/latex]
Find the GCF.[latex]3\cdot3\cdot{c}\cdot{c}\cdot{d}=9c^{2}d[/latex]
Rewrite each term as the product of the GCF and the remaining terms.[latex]\begin{array}{l}\,\,81c^{3}d=9c^{2}d\left(9c\right)\\45c^{2}d^{2}=9c^{2}d\left(5d\right)\end{array}[/latex]
Rewrite the polynomial expression using the factored terms in place of the original terms.[latex]9c^{2}d\left(9c\right)+9c^{2}d\left(5d\right)[/latex]
Factor out [latex]9c^{2}d[/latex].[latex]9c^{2}d\left(9c+5d\right)[/latex]
Answer
[latex-display]9c^{2}d\left(9c+5d\right)[/latex-display]Summary
A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\cdot{b}+a\cdot{c}[/latex] form has been rewritten as [latex]a\left(b+c\right)[/latex], where a is the GCF, the polynomial is in factored form.Contribute!
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- Question ID 146341, 146340, 146339, 146338, 146337, 146335, 146333, 146331, 146330. Authored by: Lumen Learning. License: CC BY: Attribution.
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- Ex: Factor a Binomial - Greatest Common Factor (Basic). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
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