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Mathematics LibreTexts

10.3: Multiply Polynomials (Part 2)

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    7268
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    Using the FOIL Method

    Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.

    $$\begin{split} (x + 2)&(x - y) \\ x^{2} - xy &+ 2x - 2y \end{split}$$

    Where did the first term, x2, come from?

    It is the product of x and x, the first terms in (x + 2) and (x − y).

    CNX_BMath_Figure_10_03_016_img.jpg

    The next term, −xy, is the product of x and − y, the two outer terms.

    CNX_BMath_Figure_10_03_017_img.jpg

    The third term, +2x, is the product of 2 and x, the two inner terms.

    CNX_BMath_Figure_10_03_018_img.jpg

    And the last term, −2y, came from multiplying the two last terms.

    CNX_BMath_Figure_10_03_019_img.jpg

    We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.

    CNX_BMath_Figure_10_03_025_img.jpg

    Let's look at (x + 3)(x + 7) again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.

    CNX_BMath_Figure_10_03_063_img.jpg

    Example 10.38:

    Multiply using the FOIL method: (x + 6)(x + 9).

    Solution
    Step 1: Multiply the First terms. CNX_BMath_Figure_10_03_054_img-01.png
    Step 2: Multiply the Outer terms. CNX_BMath_Figure_10_03_054_img-02.png
    Step 3: Multiply the Inner terms. CNX_BMath_Figure_10_03_054_img-03.png
    Step 4: Multiply the Last terms. CNX_BMath_Figure_10_03_054_img-04.png
    Step 5: Combine like terms, when possible. x2 + 15x + 54

    Exercise 10.75:

    Multiply using the FOIL method: (x + 7)(x + 8).

    Exercise 10.76:

    Multiply using the FOIL method: (y + 14)(y + 2).

    We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

    HOW TO: USE THE FOIL METHOD FOR MULTIPLYING TWO BINOMIALS

    Step 1. Multiply the First terms.

    Step 2. Multiply the Outer terms.

    Step 3. Multiply the Inner terms.

    Step 4. Multiply the Last terms.

    Step 5. Combine like terms, when possible.

    CNX_BMath_Figure_10_03_025_img.jpg

    Example 10.39:

    Multiply: (y − 8)(y + 6).

    Solution
    Step 1: Multiply the First terms. CNX_BMath_Figure_10_03_055_img-01.png
    Step 2: Multiply the Outer terms. CNX_BMath_Figure_10_03_055_img-02.png
    Step 3: Multiply the Inner terms. CNX_BMath_Figure_10_03_055_img-03.png
    Step 4: Multiply the Last terms. CNX_BMath_Figure_10_03_055_img-04.png
    Step 5: Combine like terms, when possible. y2 - 2y - 48

    Exercise 10.77:

    Multiply: (y − 3)(y + 8).

    Exercise 10.78:

    Multiply: (q − 4)(q + 5).

    Example 10.40:

    Multiply: (2a + 3)(3a − 1).

    Solution
      CNX_BMath_Figure_10_03_056_img-02.png
    Multiply the First terms. CNX_BMath_Figure_10_03_056_img-03.png
    Multiply the Outer terms. CNX_BMath_Figure_10_03_056_img-04.png
    Multiply the Inner terms. CNX_BMath_Figure_10_03_056_img-05.png
    Multiply the Last terms. CNX_BMath_Figure_10_03_056_img-06.png
    Combine like terms. 6a2 + 7a - 3

    Exercise 10.79:

    Multiply: (4a + 9)(5a − 2).

    Exercise 10.80:

    Multiply: (7x + 4)(7x − 8).

    Example 10.41:

    Multiply: (5x − y)(2x − 7).

    Solution
      CNX_BMath_Figure_10_03_057_img-02.png
    Multiply the First terms. CNX_BMath_Figure_10_03_057_img-03.png
    Multiply the Outer terms. CNX_BMath_Figure_10_03_057_img-04.png
    Multiply the Inner terms. CNX_BMath_Figure_10_03_057_img-05.png
    Multiply the Last terms. CNX_BMath_Figure_10_03_057_img-06.png
    Combine like terms. There are none. 10x2 - 35x - 2xy + 7y

    Exercise 10.81:

    Multiply: (12x − y)(x − 5).

    Exercise 10.82:

    Multiply: (6a − b)(2a − 9).

    Using the Vertical Method

    The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

    CNX_BMath_Figure_10_03_058_img.jpg

    You start by multiplying 23 by 6 to get 138. Then you multiply 23 by 4, lining up the partial product in the correct columns. Last, you add the partial products. Now we'll apply this same method to multiply two binomials.

    Example 10.42:

    Multiply using the vertical method: (5x − 1)(2x − 7).

    Solution

    It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied 23(46).

      CNX_BMath_Figure_10_03_059_img-01.png
    Multiply 2x − 7 by −1. CNX_BMath_Figure_10_03_059_img-02.png
    Multiply 2x − 7 by 5x. CNX_BMath_Figure_10_03_059_img-03.png
    Add like terms. CNX_BMath_Figure_10_03_059_img-04.png

    Notice the partial products are the same as the terms in the FOIL method.

    CNX_BMath_Figure_10_03_060_img.jpg

    Exercise 10.83:

    Multiply using the vertical method: (4m − 9)(3m − 7).

    Exercise 10.84:

    Multiply using the vertical method: (6n − 5)(7n − 2).

    We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.

    Definition: Multiplying Two Binomials

    To multiply binomials, use the:

    • Distributive Property
    • FOIL Method
    • Vertical Method

    Remember, FOIL only works when multiplying two binomials.

    Multiply a Trinomial by a Binomial

    We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

    Example 10.43:

    Multiply using the Distributive Property: (x + 3)(2x2 − 5x + 8).

    Solution
      CNX_BMath_Figure_10_03_061_img-01.png
    Distribute. CNX_BMath_Figure_10_03_061_img-02.png
    Multiply. 2x3 − 5x2 + 8x + 6x2 − 15x + 24
    Combine like terms. 2x3 + x2 − 7x + 24

    Exercise 10.85:

    Multiply using the Distributive Property: (y − 1)(y2 − 7y + 2).

    Exercise 10.86:

    Multiply using the Distributive Property: (x + 2)(3x2 − 4x + 5).

    Now let's do this same multiplication using the Vertical Method.

    Example 10.44:

    Multiply using the Vertical Method: (x + 3)(2x2 − 5x + 8).

    Solution

    It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

      CNX_BMath_Figure_10_03_062_img-01.png
    Multiply (2x2 − 5x + 8) by 3. CNX_BMath_Figure_10_03_062_img-02.png
    Multiply (2x2 − 5x + 8) by x. CNX_BMath_Figure_10_03_062_img-03.png
    Add like terms. CNX_BMath_Figure_10_03_062_img-04.png

    Exercise 10.87:

    Multiply using the Vertical Method: (y − 1)(y2 − 7y + 2).

    Exercise 10.88:

    Multiply using the Vertical Method: (x + 2)(3x2 − 4x + 5).

    Practice Makes Perfect

    Multiply a Polynomial by a Monomial

    In the following exercises, multiply.

    1. 4(x + 10)
    2. 6(y + 8)
    3. 15(r − 24)
    4. 12(v − 30)
    5. −3(m + 11)
    6. −4(p + 15)
    7. −8(z − 5)
    8. −3(x − 9)
    9. u(u + 5)
    10. q(q + 7)
    11. n(n2 − 3n)
    12. s(s2 − 6s)
    13. 12x(x − 10)
    14. 9m(m − 11)
    15. −9a(3a + 5)
    16. −4p(2p + 7)
    17. 6x(4x + y)
    18. 5a(9a + b)
    19. 5p(11p − 5q)
    20. 12u(3u − 4v)
    21. 3(v2 + 10v + 25)
    22. 6(x2 + 8x + 16)
    23. 2n(4n2 − 4n + 1)
    24. 3r(2r2 − 6r + 2)
    25. −8y(y2 + 2y − 15)
    26. −5m(m2 + 3m − 18)
    27. 5q3(q2 − 2q + 6)
    28. 9r3(r2 − 3r + 5)
    29. −4z2(3z2 + 12z − 1)
    30. −3x2(7x2 + 10x − 1)
    31. (2y − 9)y
    32. (8b − 1)b
    33. (w − 6) • 8
    34. (k − 4) • 5

    Multiply a Binomial by a Binomial

    In the following exercises, multiply the following binomials using: (a) the Distributive Property (b) the FOIL method (c) the Vertical method

    1. (x + 4)(x + 6)
    2. (u + 8)(u + 2)
    3. (n + 12)(n − 3)
    4. (y + 3)(y − 9)

    In the following exercises, multiply the following binomials. Use any method.

    1. (y + 8)(y + 3)
    2. (x + 5)(x + 9)
    3. (a + 6)(a + 16)
    4. (q + 8)(q + 12)
    5. (u − 5)(u − 9)
    6. (r − 6)(r − 2)
    7. (z − 10)(z − 22)
    8. (b − 5)(b − 24)
    9. (x − 4)(x + 7)
    10. (s − 3)(s + 8)
    11. (v + 12)(v − 5)
    12. (d + 15)(d − 4)
    13. (6n + 5)(n + 1)
    14. (7y + 1)(y + 3)
    15. (2m − 9)(10m + 1)
    16. (5r − 4)(12r + 1)
    17. (4c − 1)(4c + 1)
    18. (8n − 1)(8n + 1)
    19. (3u − 8)(5u − 14)
    20. (2q − 5)(7q − 11)
    21. (a + b)(2a + 3b)
    22. (r + s)(3r + 2s)
    23. (5x − y)(x − 4)
    24. (4z − y)(z − 6)

    Multiply a Trinomial by a Binomial

    In the following exercises, multiply using (a) the Distributive Property and (b) the Vertical Method.

    1. (u + 4)(u2 + 3u + 2)
    2. (x + 5)(x2 + 8x + 3)
    3. (a + 10)(3a2 + a − 5)
    4. (n + 8)(4n2 + n − 7)

    In the following exercises, multiply. Use either method.

    1. (y − 6)(y2 − 10y + 9)
    2. (k − 3)(k2 − 8k + 7)
    3. (2x + 1)(x2 − 5x − 6)
    4. (5v + 1)(v2 − 6v − 10)

    Everyday Math

    1. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15. Think of 13 as 10 + 3 and 15 as 10 + 5.
      1. Multiply (10 + 3)(10 + 5) by the FOIL method.
      2. Multiply 13 • 15 without using a calculator.
      3. Which way is easier for you? Why?
    2. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as 20 − 2 and 17 as 20 − 3.
      1. Multiply (20 − 2)(20 − 3) by the FOIL method.
      2. Multiply 18 • 17 without using a calculator.
      3. Which way is easier for you? Why?

    Writing Exercises

    1. Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why?
    2. Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?

    Self Check

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

    CNX_BMath_Figure_AppB_061.jpg

    (b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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