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Mathematics LibreTexts

10.6: Introduction to Factoring Polynomials

  • Page ID
    5017
  • [ "article:topic", "authorname:openstax" ]

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    Skills to Develop

    • Find the greatest common factor of two or more expressions
    • Factor the greatest common factor from a polynomial

    be prepared!

    Before you get started, take this readiness quiz.

    1. Factor 56 into primes. If you missed this problem, review Example 2.48.
    2. Multiply: −3(6a + 11). If you missed this problem, review Example 7.25.
    3. Multiply: 4x2(x2 + 3x − 1). If you missed this problem, review Example 10.32.

    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.

    CNX_BMath_Figure_10_06_001_img.jpg

    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.

    Definition: Greatest Common Factor

    The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

    First we will find the greatest common factor of two numbers

    Example 10.80:

    Find the greatest common factor of 24 and 36.

    Solution
    Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. Factor 24 and 36. CNX_BMath_Figure_10_06_024_img-01.png
    Step 2: List all factors—matching common factors in a column.   CNX_BMath_Figure_10_06_024_img-02.png
    In each column, circle the common factors. Circle the 2, 2, and 3 that are shared by both numbers. CNX_BMath_Figure_10_06_024_img-03.png
    Step 3: Bring down the common factors that all expressions share. Bring down the 2, 2, 3 and then multiply.  
    Step 4: Multiply the factors.   The GCF of 24 and 36 is 12.

    Notice that since the GCF is a factor of both numbers, 24 and 36 can be written as multiples of 12.

    $$\begin{split} 24 &= 12 \cdot 2 \\ 36 &= 12 \cdot 3 \end{split}$$

    Exercise 10.159:

    Find the greatest common factor: 54, 36.

    Exercise 10.160:

    Find the greatest common factor: 48, 80.

    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.

    HOW TO: FIND THE GREATEST COMMON FACTOR

    Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.

    Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.

    Step 3. Bring down the common factors that all expressions share.

    Step 4. Multiply the factors.

    Example 10.81:

    Find the greatest common factor of 5x and 15.

    Solution

    Factor each number into primes.

    Circle the common factors in each column.

    Bring down the common factors.

    CNX_BMath_Figure_10_06_025_img-01.png

    The GCF of 5x and 15 is 5.

    Exercise 10.161:

    Find the greatest common factor: 7y, 14.

    Exercise 10.162:

    Find the greatest common factor: 22, 11m.

    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:

    Find the greatest common factor of 12x2 and 18x3.

    Solution

    Factor each coefficient into primes and write the variables with exponents in expanded form.

    Circle the common factors in each column.

    Bring down the common factors.

    Multiply the factors.

    CNX_BMath_Figure_10_06_026_img-01.png

    The GCF of 12x2 and 18x3 is 6x2.

    Exercise 10.163:

    Find the greatest common factor: 16x2, 24x3.

    Exercise 10.164:

    Find the greatest common factor: 27y3, 18y4.

    Example 10.83:

    Find the greatest common factor of 14x3, 8x2, 10x.

    Solution

    Factor each coefficient into primes and write the variables with exponents in expanded form.

    Circle the common factors in each column.

    Bring down the common factors.

    Multiply the factors.

    CNX_BMath_Figure_10_06_027_img-01.png

    The GCF of 14x3 and 8x2, and 10x is 2x.

    Exercise 10.165:

    Find the greatest common factor: 21x3, 9x2, 15x.

    Exercise 10.166:

    Find the greatest common factor: 25m4, 35m3, 20m2.

    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 • 6 or 3 • 4), 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:

    $$\begin{split} 2(x &+ 7) \quad factors \\ 2 \cdot x &+ 2 \cdot 7 \\ 2x &+ 14 \quad product \end{split}$$

    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”.

    Definition: Distributive Property

    If a, b, c are real numbers, then a(b + c) = ab + ac and ab + ac = a(b + c).

    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:

    Factor: 2x + 14.

    Solution
    Step 1: Find the GCF of all the terms of the polynomial. Find the GCF of 2x and 14. CNX_BMath_Figure_10_06_028_img-01.png
    Step 2: Rewrite each term as a product using the GCF. Rewrite 2x and 14 as products of their GCF, 2.$$\begin{split} 2x &= 2 \cdot x \\ 14 &= 2 \cdot 7 \end{split}$$ $$\begin{split} 2x &+ 14 \\ \textcolor{red}{2} \cdot x &+ \textcolor{red}{2} \cdot 7 \end{split}$$
    Step 3: Use the Distributive Property 'in reverse' to factor the expression.   2(x + 7)
    Step 4: Check by multiplying the factors. Check. $$\begin{split} 2(x &+ 7) \\ 2 \cdot x &+ 2 \cdot 7 \\ 2x &+ 14\; \checkmark \end{split}$$

    Exercise 10.167:

    Factor: 4x + 12.

    Exercise 10.168:

    Factor: 6a + 24.

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

    Noun 7 is a factor of 14
    Verb Factor 2 from 2x + 14

    HOW TO: FACTOR THE GREATEST COMMON FACTOR FROM A POLYNOMIAL

    Step 1. Find the GCF of all the terms of the polynomial.

    Step 2. Rewrite each term as a product using the GCF.

    Step 3. Use the Distributive Property ‘in reverse’ to factor the expression.

    Step 4. Check by multiplying the factors.

    Example 10.85:

    Factor: 3a + 3.

    Solution
    CNX_BMath_Figure_10_06_029_img-01.png
    Rewrite each term as a product using the GCF. $$\textcolor{red}{3} \cdot a + \textcolor{red}{3} \cdot 1$$
    Use the Distributive Property 'in reverse' to factor the GCF. $$3(a+1)$$
    Check by multiplying the factors to get the original polynomial. $$\begin{split} 3(a &+ 1) \\ 3 \cdot a &= 3 \cdot 1 \\ 3a &+ 3\; \checkmark \end{split}$$

    Exercise 10.169:

    Factor: 9a + 9.

    Exercise 10.170:

    Factor: 11x + 11.

    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:

    Factor: 12x − 60.

    CNX_BMath_Figure_10_06_030_img-01.png
    Rewrite each term as a product using the GCF. $$\textcolor{red}{12} \cdot x - \textcolor{red}{12} \cdot 5$$
    Factor the GCF. $$12(x-5)$$
    Check by multiplying the factors. $$\begin{split} 12(x &- 5) \\ 12 \cdot x &- 12 \cdot 5 \\ 12x &- 60\; \checkmark \end{split}$$

    Exercise 10.171:

    Factor: 11x − 44.

    Exercise 10.172:

    Factor: 13y − 52.

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

    Example 10.87:

    Factor: 3y2 + 6y + 9.

    Solution
    CNX_BMath_Figure_10_06_031_img-01.png
    Rewrite each term as a product using the GCF. $$\textcolor{red}{3} \cdot y^{2} + \textcolor{red}{3} \cdot 2y + \textcolor{red}{3} \cdot 3$$
    Factor the GCF. $$3(y^{2} + 2y + 3)$$
    Check by multiplying. $$\begin{split} 3(y^{2} &+ 2y + 3) \\ 3 \cdot y^{2} &+ 3 \cdot 2y + 3 \cdot 3 \\ 3y^{2} &+ 6y + 9\; \checkmark \end{split}$$

    Exercise 10.173:

    Factor: 4y2 + 8y + 12.

    Exercise 10.174:

    Factor: 6x2 + 42x − 12.

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

    Example 10.88:

    Factor: 6x2 + 5x.

    Solution
    Find the GCF of 6x2 and 5x and the math that goes with it. CNX_BMath_Figure_10_06_013_img-1.jpg
    Rewrite each term as a product. $$\textcolor{red}{x} \cdot 6x + \textcolor{red}{x} \cdot 5$$
    Factor the GCF. $$x(6x + 5)$$
    Check by multiplying. $$\begin{split} x(6x &+ 5) \\ x \cdot 6x &+ x \cdot 5 \\ 6x^{2} &+ 5x\; \checkmark \end{split}$$

    Exercise 10.175:

    Factor: 9x2 + 7x.

    Exercise 10.176:

    Factor: 5a2 − 12a.

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

    Example 10.89:

    Factor: 4x3 − 20x2.

    Solution
    CNX_BMath_Figure_10_06_033_img-01.png
    Rewrite each term. $$\textcolor{red}{4x^{2}} \cdot x - \textcolor{red}{4x^{2}} \cdot 5$$
    Factor the GCF. $$4x^{2} (x-5)$$
    Check. $$\begin{split} 4x^{2} (x &- 5) \\ 4x^{2} \cdot x &- 4x^{2} \cdot 5 \\ 4x^{3} &- 20x^{2}\; \checkmark \end{split}$$

    Exercise 10.177:

    Factor: 2x3 + 12x2.

    Exercise 10.178:

    Factor: 6y− 15y2.

    Example 10.90:

    Factor: 21y2 + 35y.

    Solution
    Find the GCF of 21y2 and 35y. CNX_BMath_Figure_10_06_034_img-01.png
    Rewrite each term. $$\textcolor{red}{7y} \cdot 3y + \textcolor{red}{7y} \cdot 5$$
    Factor the GCF. $$7y(3y + 5)$$

    Exercise 10.179:

    Factor: 18y2 + 63y.

    Exercise 10.180:

    Factor: 32k2 + 56k.

    Example 10.91:

    Factor: 14x3 + 8x2 − 10x.

    Solution

    Previously, we found the GCF of 14x3, 8x2, and 10x to be 2x.

    Rewrite each term using the GCF, 2x. $$\textcolor{red}{2x} \cdot 7x^{2} + \textcolor{red}{2x} \cdot 4x - \textcolor{red}{2x} \cdot 5$$
    Factor the GCF. $$2x(7x^{2} + 4x - 5)$$
    Check. $$\begin{split} 2x(7x^{2} &+ 4x - 5) \\ 2x \cdot 7x^{2} &+ 2x \cdot 4x - 2x \cdot 5 \\ 14x^{3} &+ 8x^{2} - 10x\; \checkmark \end{split}$$

    Exercise 10.181:

    Factor: 18y3 − 6y2 − 24y.

    Exercise 10.182:

    Factor: 16x3 + 8x2 − 12x.

    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:

    Factor: −9y − 27.

    Solution
    When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9y and 27 is 9. CNX_BMath_Figure_10_06_036_img-01.png
    Since the expression −9y − 27 has a negative leading coefficient, we use −9 as the GCF.  
    Rewrite each term using the GCF. $$\textcolor{red}{-9} \cdot y + (\textcolor{red}{-9}) \cdot 3$$
    Factor the GCF. $$-9(y+3)$$
    Check. $$\begin{split} -9(y &+ 3) \\ -9 \cdot y &+ (-9) \cdot 3 \\ -9y &- 27\; \checkmark \end{split}$$

    Exercise 10.183:

    Factor: −5y − 35.

    Exercise 10.184:

    Factor: −16z − 56.

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

    Example 10.93:

    Factor: −4a2 + 16a.

    Solution
    The leading coefficient is negative, so the GCF will be negative. CNX_BMath_Figure_10_06_037_img-01.png
    Since the leading coefficient is negative, the GCF is negative, −4a.  
    Rewrite each term. $$\textcolor{red}{-4a} \cdot a - (\textcolor{red}{-4a}) \cdot 4$$
    Factor the GCF. $$-4a(a-4)$$
    Check on your own by multiplying.  

    Exercise 10.185:

    Factor: −7a2 + 21a.

    Exercise 10.186:

    Factor: −6x2 + x.

    ACCESS ADDITIONAL ONLINE RESOURCES

    Factor GCF

    Factor a Binomial

    Identify GCF

    Practice Makes Perfect

    Find the Greatest Common Factor of Two or More Expressions

    In the following exercises, find the greatest common factor.

    1. 40, 56
    2. 45, 75
    3. 72, 162
    4. 150, 275
    5. 3x, 12
    6. 4y, 28
    7. 10a, 50
    8. 5b, 30
    9. 16y, 24y2
    10. 9x, 15x2
    11. 18m3, 36m2
    12. 12p4, 48p3
    13. 10x, 25x2, 15x3
    14. 18a, 6a2, 22a3
    15. 24u, 6u2, 30u3
    16. 40y, 10y2, 90y3
    17. 15a4, 9a5, 21a6
    18. 35x3, 10x4, 5x5
    19. 27y2, 45y3, 9y4
    20. 14b2, 35b3, 63b4

    Factor the Greatest Common Factor from a Polynomial

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

    1. 2x + 8
    2. 5y + 15
    3. 3a − 24
    4. 4b − 20
    5. 9y − 9
    6. 7x − 7
    7. 5m2 + 20m + 35
    8. 3n+ 21n + 12
    9. 8p2 + 32p + 48
    10. 6q2 + 30q + 42
    11. 8q2 + 15q
    12. 9c2 + 22c
    13. 13k2 + 5k
    14. 17x2 + 7x
    15. 5c2 + 9c
    16. 4q2 + 7q
    17. 5p2 + 25p
    18. 3r2 + 27r
    19. 24q2 − 12q
    20. 30u2 − 10u
    21. yz + 4z
    22. ab + 8b
    23. 60x − 6x3
    24. 55y − 11y4
    25. 48r4 − 12r3
    26. 45c3 − 15c2
    27. 4a3 − 4ab2
    28. 6c3 − 6cd2
    29. 30u3 + 80u2
    30. 48x3 + 72x2
    31. 120y6 + 48y4
    32. 144a6 + 90a3
    33. 4q2 + 24q + 28
    34. 10y2 + 50y + 40
    35. 15z2 − 30z − 90
    36. 12u2 − 36u − 108
    37. 3a4 − 24a3 + 18a2
    38. 5p4 − 20p3 − 15p2
    39. 11x6 + 44x5 − 121x4
    40. 8c5 + 40c4 − 56c3
    41. −3n − 24
    42. −7p − 84
    43. −15a2 − 40a
    44. −18b2 − 66b
    45. −10y3 + 60y2
    46. −8a3 + 32a2
    47. −4u5 + 56u3
    48. −9b5 + 63b3

    Everyday Math

    1. Revenue A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of p dollars each is given by the polynomial −5p2 + 150p. Factor the greatest common factor from this polynomial.
    2. Height of a baseball The height of a baseball hit with velocity 80 feet/second at 4 feet above ground level is −16t2 + 80t + 4, with t = the number of seconds since it was hit. Factor the greatest common factor from this polynomial.

    Writing Exercises

    1. The greatest common factor of 36 and 60 is 12. Explain what this means.
    2. What is the GCF of y4, y5, and y10? Write a general rule that tells how to find the GCF of ya, yb, and yc.

    Self Check

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

    CNX_BMath_Figure_AppB_065.jpg

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

    Contributors