10.6 Introduction to Factoring Polynomials

Learning objectives

Learning objectives

• Find the greatest common factor of two or more expressions

• Factor the greatest common factor from a polynomial

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

First we will find the greatest common factor of two numbers.

Example 10.80

Solution

1. Factor both numbers into primes. The prime factorizations are $24=2\cdot{}2\cdot{}2\cdot{}3$ and $36=2\cdot{}2\cdot{}3\cdot{}3$.
2. Identify the common prime factors. Both factorizations contain two factors of $2$ and one factor of $3$.
3. Multiply the common factors. The common product is $2\cdot{}2\cdot{}3=12$.
4. Verify the greatest common factor. Since $24=12\cdot{}2$ and $36=12\cdot{}3$, the greatest common factor of $24$ and $36$ is $12$.

Takeaway: Since $24=12\cdot2$ and $36=12\cdot3$, the greatest common factor of $24$ and $36$ is $12$.

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

Example 10.81

Solution

1. Factor each term. Write $5x=5\cdot x$ and $15=3\cdot5$, so the common numerical factor is visible in both terms.
2. Choose the greatest common factor. The only factor shared by both factored terms is $5$, which the figure circles. The variable $x$ is not a factor of $15$, so it cannot be included.
3. State the result. The greatest common factor of $5x$ and $15$ is $5$.

Takeaway: A common factor must appear in every term, so a variable appears in the GCF only if it occurs in all terms.

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

Example 10.82

Solution

1. Factor the numerical coefficients. Write $12x^2=2^2\cdot3\cdot x^2$ and $18x^3=2\cdot3^2\cdot x^3$, so the greatest common numerical factor is $2\cdot3=6$.
2. Compare the variable powers. Both terms contain at least $x^2$, but only the second contains $x^3$, so the greatest common variable power is $x^2$.
3. Combine the common factors. The figure circles the common numerical factors and $x^2$, giving the greatest common factor $6x^2$.
4. State the result. The greatest common factor of $12x^2$ and $18x^3$ is $6x^2$.

Takeaway: The GCF of monomials combines the greatest common numerical factor with the smallest shared exponent of each variable.

Example 10.83

Solution

1. Factor the coefficients. Write $14x^3=2\cdot7\cdot x^3$, $8x^2=2^3\cdot x^2$, and $10x=2\cdot5\cdot x$, as displayed in the figure.
2. Find the greatest common numerical factor. The only common numerical factor of all three coefficients is $2$.
3. Find the greatest common variable power. The variable exponents are $3$, $2$, and $1$, so the smallest exponent present in every term is $1$.
4. Combine and state the result. The circled common factors in the figure multiply to give the greatest common factor $2x$.

Takeaway: For several terms, the GCF contains only the numerical factors and variable powers common to every term.

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, $12$ as $2\cdot{}6\,\mathrm{or}\,3\cdot{}4\mathrm{),}$ 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:

Here, we will start with a product, like $2x+14,$ and end with its factors, $2(x+7).$ To do this we apply the Distributive Property “in reverse”.

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 10.84

Solution

1. Find the greatest common factor. The terms $2x$ and $14$ share the factor $2$, as the first figure shows by factoring $14$ as $2\cdot7$.
2. Rewrite each term using the common factor. Write $2x+14=2x+2\cdot7$, which is the second figure's factorization of the two terms.
3. Factor out the greatest common factor. Apply the distributive property in reverse to obtain $2(x+7)$.
4. Check by distributing. Multiplying $2$ by each term inside the parentheses gives $2x+14$, exactly the original expression, as the third figure confirms.

Takeaway: Factoring the greatest common factor means writing every term as a product with that factor and then reversing the distributive property.

Notice that in Example 10.84, we used the word factor as both a noun and a verb:

Example 10.85

Solution

1. Find the greatest common factor. The terms $3a$ and $3$ have the common numerical factor $3$, while $a$ is not a factor of the second term. The first figure therefore identifies the GCF as $3$.
2. Rewrite both terms. Use the GCF to write $3a+3=3a+3\cdot1$, which is shown in the third figure after the original expression in the second figure.
3. Factor out the common factor. Reverse the distributive property to write $3(a+1)$.
4. Check the result. Distributing $3$ gives $3a+3$, so the original expression is correctly factored as $3(a+1)$.

Takeaway: When a term is only the GCF, rewrite it as the GCF times $1$ before factoring.

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 10.86

Solution

1. Find the greatest common factor. Prime factorizations give $12x=2^2\cdot3\cdot x$ and $60=2^2\cdot3\cdot5$, so the shared factor is $12$. The first figure circles the common factors.
2. Rewrite each term. Factoring out $12$ gives $12x-60=12x-12\cdot5$, which appears in the third figure after the original expression in the second.
3. Factor out the greatest common factor. Reverse the distributive property to obtain $12(x-5)$.
4. Check by distributing. Multiplying $12$ by $x$ and by $-5$ gives $12x-60$, confirming the factored form shown in the final figure.

Takeaway: The greatest common factor must include every numerical and variable factor common to all terms.

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Example 10.87

Solution

1. Find the greatest common factor. The prime factorizations of $3y^2$, $6y$, and $9$ all contain $3$, but the variable powers are $2$, $1$, and $0$. Their smallest common power is therefore zero, so the GCF is $3$, as the first figure shows.
2. Rewrite each term. Use the GCF to write $3y^2+6y+9=3y^2+3\cdot2y+3\cdot3$, with the repeated factor highlighted in the third figure.
3. Factor out the greatest common factor. Reverse the distributive property to obtain $3(y^2+2y+3)$.
4. Check by distributing. Multiplying $3$ by all three terms inside the parentheses gives $3y^2+6y+9$, so the final figure confirms the result.

Takeaway: For the GCF of several terms, use the smallest common power of each variable, even when that power is zero.

In the next example, we factor a variable from a binomial.

Example 10.88

Solution

1. Find the greatest common factor. The terms $6x^2$ and $5x$ share the variable factor $x$, but their numerical coefficients have no common factor other than $1$. The first figure highlights $x$.
2. Rewrite each term. Factor $x$ from both terms to get $x\cdot6x+x\cdot5$, which the second figure displays with both copies of $x$ highlighted.
3. Factor out the common factor. Reverse the distributive property to obtain $x(6x+5)$.
4. State the factored form. Thus $6x^2+5x=x(6x+5)$.

Takeaway: When no numerical factor is common, the GCF may consist only of a variable factor.

When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

Example 10.89

Solution

1. Find the greatest common factor. The coefficients $4$ and $20$ share $4$, and the variable exponents $3$ and $2$ share $x^2$. The first figure circles the common factors $4$ and $x^2$, giving the GCF $4x^2$.
2. Rewrite each term. Factor out $4x^2$ to obtain $4x^3-20x^2=4x^2\cdot x-4x^2\cdot5$, as shown in the third figure after the original expression.
3. Factor out the common factor. Reverse the distributive property to write $4x^2(x-5)$.
4. Check by distributing. Multiplying gives $4x^3-20x^2$, confirming the factored form shown in the final figure.

Takeaway: When factoring a binomial, use the smaller variable exponent and the greatest common numerical factor.

Example 10.90

Solution

1. Find the greatest common factor. The prime factorizations of $21y^2$ and $35y$ share $7$ and $y$, so the GCF is $7y$, as highlighted in the first figure.
2. Rewrite each term. Factor out $7y$ to get $21y^2+35y=7y\cdot3y+7y\cdot5$, which is displayed in the third figure after the original expression.
3. Factor out the common factor. Reverse the distributive property to write $7y(3y+5)$.
4. Check by distributing. Multiplying $7y$ by $3y+5$ gives $21y^2+35y$, confirming the factored form in the final figure.

Takeaway: A common variable factor uses the smaller exponent, while a common numerical factor uses the greatest shared divisor.

Example 10.91

Solution

1. Identify the common factor. The terms $14x^3$, $8x^2$, and $-10x$ all contain $2x$: the coefficients share $2$, and the smallest common variable exponent is $1$.
2. Rewrite every term. Factoring out $2x$ gives $2x\cdot7x^2+2x\cdot4x-2x\cdot5$, as the first figure shows.
3. Factor out the greatest common factor. Reverse the distributive property to obtain $2x(7x^2+4x-5)$.
4. Check by distributing. The products are $14x^3+8x^2-10x$, so the second figure confirms that the original trinomial is correctly factored.

Takeaway: A trinomial's GCF uses the smallest common numerical factor and the smallest common variable power.

When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.

Example 10.92

Solution

1. Find the magnitude of the greatest common factor. Since $9y=3\cdot3\cdot y$ and $27=3\cdot3\cdot3$, the common numerical factor is $9$. The first figure circles the two shared factors of $3$.
2. Place the negative sign on the factor. The terms are $-9y$ and $-27$, so write $-9y=(-9)y$ and $-27=(-9)\cdot3$, matching the second figure.
3. Factor out the common factor. Reverse the distributive property to obtain $-9(y+3)$.
4. Check the signs. Distributing $-9$ gives $-9y-27$, so the final figure confirms the factored form $-9(y+3)$.

Takeaway: A negative common factor must multiply every term inside the parentheses so that the original signs are preserved.

Pay close attention to the signs of the terms in the next example.

Example 10.93

Solution

1. Find the greatest common factor. The coefficients $4$ and $16$ share $4$, and the variable exponents $2$ and $1$ share $a$, so the GCF has magnitude $4a$. The first figure circles the common factors $2$, $2$, and $a$.
2. Place the negative sign on the common factor. Rewrite the terms as $-4a^2=(-4a)a$ and $16a=(-4a)(-4)$, matching the second figure.
3. Factor out the complete common factor. Reverse the distributive property to obtain $-4a(a-4)$.
4. Check the signs. Distributing $-4a$ gives $-4a^2+16a$, so the original expression is correctly factored as $-4a(a-4)$.

Takeaway: Put the sign on the extracted GCF and adjust the sign inside the parentheses so the original terms are preserved.

Section 10.6 Exercises

Section 10.6 Exercises

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

1. Find the greatest common factor: $16{x}^{2},\,24{x}^{3}.$

2. Find the greatest common factor: $27{y}^{3},\,18{y}^{4}.$

3. Find the greatest common factor: $21{x}^{3},\,9{x}^{2},\,15x.$

4. Find the greatest common factor: $25{m}^{4},\,35{m}^{3},\,20{m}^{2}.$

1. Factor: $4x+12.$

2. Factor: $6a+24.$

3. Factor: $9a+9.$

4. Factor: $11x+11.$

1. Factor: $11x-44.$

2. Factor: $13y-52.$

3. Factor: $4{y}^{2}+8y+12.$

4. Factor: $6{x}^{2}+42x-12.$

1. Factor: $9{x}^{2}+7x.$

2. Factor: $5{a}^{2}-12a.$

3. Factor: $2{x}^{3}+12{x}^{2}.$

4. Factor: $6{y}^{3}-15{y}^{2}.$

1. Factor: $18{y}^{2}+63y.$

2. Factor: $32{k}^{2}+56k.$

3. Factor: $18{y}^{3}-6{y}^{2}-24y.$

4. Factor: $16{x}^{3}+8{x}^{2}-12x.$

1. Factor: $-5y-35.$

2. Factor: $-16z-56.$

3. Factor: $-7{a}^{2}+21a.$

4. Factor: $-6{x}^{2}+x.$

1. $9x,\,15{x}^{2}$

2. $12{p}^{4},\,48{p}^{3}$

3. $18a,\,6{a}^{2},\,22{a}^{3}$

4. $40y,\,10{y}^{2},\,90{y}^{3}$

1. $35{x}^{3},\,10{x}^{4},\,5{x}^{5}$

2. $14{b}^{2},\,35{b}^{3},\,63{b}^{4}$

3. $5y+15$

4. $4b-20$

1. $7x-7$

2. $3{n}^{2}+21n+12$

3. $6{q}^{2}+30q+42$

4. $9{c}^{2}+22c$

1. $17{x}^{2}+7x$

2. $4{q}^{2}+7q$

3. $3{r}^{2}+27r$

4. $30{u}^{2}-10u$

1. $ab+8b$

2. $55y-11{y}^{4}$

3. $45{c}^{3}-15{c}^{2}$

4. $6{c}^{3}-6c{d}^{2}$

1. $48{x}^{3}+72{x}^{2}$

2. $144{a}^{6}+90{a}^{3}$

3. $10{y}^{2}+50y+40$

4. $12{u}^{2}-36u-108$

1. $5{p}^{4}-20{p}^{3}-15{p}^{2}$

2. $8{c}^{5}+40{c}^{4}-56{c}^{3}$

3. $-7p-84$

4. $-18{b}^{2}-66b$

1. $-8{a}^{3}+32{a}^{2}$

2. $-9{b}^{5}+63{b}^{3}$

3. Height of a baseball The height of a baseball hit with velocity $80$ feet/second at $4$ feet above ground level is $-16{t}^{2}+80t+4,$ with $t=$ the number of seconds since it was hit. Factor the greatest common factor from this polynomial.

4. What is the GCF of ${y}^{4}$, ${y}^{5}$, and ${y}^{10}$? Write a general rule that tells how to find the GCF of ${y}^{a}$, ${y}^{b}$, and ${y}^{c}$.