Learning Outcomes
- Factor the greatest common factor from a polynomial
Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, [latex]12[/latex] as [latex]2\cdot 6\text{ or }3\cdot 4[/latex]) in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:
[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”.
Distributive Property
If [latex]a,b,c[/latex] are real numbers, then
[latex]a\left(b+c\right)=ab+ac\text{ and }ab+ac=a\left(b+c\right)[/latex]
The form on the left is used to multiply. The form on the right is used to factor.
So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!
example
Factor: [latex]2x+14[/latex]
Solution
Step 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]2x=2\cdot x[/latex] [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]2(x+7)[/latex] [latex]2\cdot{x}+2\cdot{7}[/latex] [latex]2x+14\quad\checkmark[/latex] |
try it
Notice that in the example, we used the word factor as both a noun and a verb:
[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]
try it
The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.
example
Factor: [latex]12x - 60[/latex]
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Watch the following video to see more examples of factoring the GCF from a binomial.
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
example
Factor: [latex]3{y}^{2}+6y+9[/latex]
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In the next example, we factor a variable from a binomial.
example
Factor: [latex]6{x}^{2}+5x[/latex]
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When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!
example
Factor: [latex]4{x}^{3}-20{x}^{2}[/latex]
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example
Factor: [latex]21{y}^{2}+35y[/latex]
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example
Factor: [latex]14{x}^{3}+8{x}^{2}-10x[/latex]
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When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.
example
Factor: [latex]-9y - 27[/latex]
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Pay close attention to the signs of the terms in the next example.
example
Factor: [latex]-4{a}^{2}+16a[/latex]