Elementary Algebra 2e · free preview

6.3 Multiply Polynomials

Learning objectives: Multiply a polynomial by a monomial; Multiply a binomial by a binomial; Multiply a trinomial by a binomial

1. Learning objectives

Learning objectives

Learning objectives

  • Multiply a polynomial by a monomial
  • Multiply a binomial by a binomial
  • Multiply a trinomial by a binomial

2. Multiply a Polynomial by a Monomial

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like 2(x−3)2(x-3). You multiplied both terms in the parentheses, x and 3x\,\mathrm{and}\,3, by 2, to get 2x−62x-6. With this chapter’s new vocabulary, you can say you were multiplying a binomial, x−3x-3, by a monomial, 2.

Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

3. Example 6.28

Example 6.28

Example 6.28

Multiply: 4(x+3).4(x+3).

Solution

4. Example 6.29

Example 6.29

Example 6.29

Multiply: y(y−2).y(y-2).

Solution

5. Example 6.30

Example 6.30

Example 6.30

Multiply: 7x(2x+y).7x(2x+y).

Solution

6. Example 6.31

Example 6.31

Example 6.31

Multiply: −2y(4y2+3y−5).-2y(4{y}^{2}+3y-5).

Solution

7. Example 6.32

Example 6.32

Example 6.32

Multiply: 2x3(x2−8x+1).2{x}^{3}({x}^{2}-8x+1).

Solution

8. Example 6.33

Example 6.33

Example 6.33

Multiply: (x+3)p.(x+3)p.

Solution

9. Multiply a Binomial by a Binomial

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

10. Multiply a Binomial by a Binomial Using the Distributive Property

Multiply a Binomial by a Binomial Using the Distributive Property

Look at Example 6.33, where we multiplied a binomial by a monomial.

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

11. Example 6.34

Example 6.34

Example 6.34

Multiply: (y+5)(y+8).(y+5)(y+8).

Solution

12. Example 6.35

Example 6.35

Example 6.35

Multiply: (2y+5)(3y+4).(2y+5)(3y+4).

Solution

13. Example 6.36

Example 6.36

Example 6.36

Multiply: (4y+3)(2y−5).(4y+3)(2y-5).

Solution

14. Example 6.37

Example 6.37

Example 6.37

Multiply: (x−2)(x−y).(x-2)(x-y).

Solution

15. Multiply a Binomial by a Binomial Using the FOIL Method

Multiply a Binomial by a Binomial 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, like in Example 6.37, 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.

(x−2)(x−y)x2−xy−2x+2y\begin{array}{c} (x-2)(x-y) \\ {x}^{2}-xy-2x+2y \\ \end{array}

Where did the first term, x2{x}^{2}, come from?

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.

(x−2)(x−y)x2−xy−2x+2yF O I L\begin{array}{c} (x-2)(x-y) \\ \underset{F\,O\,I\,L}{{x}^{2}-xy-2x+2y} \\ \end{array}

Let’s look at (x+3)(x+7)(x+3)(x+7).

Notice how the terms in third line fit the FOIL pattern.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

16. Example 6.38

Example 6.38

Example 6.38

How to Multiply a Binomial by a Binomial using the FOIL Method

Multiply using the FOIL method: (x+5)(x+9).(x+5)(x+9).

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

Multiply two binomials using the FOIL method

When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

17. Example 6.39

Example 6.39

Example 6.39

Multiply: (y−7)(y+4).(y-7)(y+4).

18. Example 6.40

Example 6.40

Example 6.40

Multiply: (4x+3)(2x−5).(4x+3)(2x-5).

The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.

19. Example 6.41

Example 6.41

Example 6.41

Multiply: (3x−y)(2x−5).(3x-y)(2x-5).

Solution

Be careful of the exponents in the next example.

20. Example 6.42

Example 6.42

Example 6.42

Multiply: (n2+4)(n−1).({n}^{2}+4)(n-1).

Solution

21. Example 6.43

Example 6.43

Example 6.43

Multiply: (3pq+5)(6pq−11).(3pq+5)(6pq-11).

Solution

22. Multiply a Binomial by a Binomial Using the Vertical Method

Multiply a Binomial by a Binomial Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it only works 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.

Now we’ll apply this same method to multiply two binomials.

23. Example 6.44

Example 6.44

Example 6.44

Multiply using the Vertical Method: (3y−1)(2y−6).(3y-1)(2y-6).

Solution

It does not matter which binomial goes on the top.

Multiply 3y−1 by −6.Multiply 3y−1 by2y.Addliketerms.3y−1 × 2y−6________−18y+66y2−2y_____________6y2−20y+6partialproductpartialproductproduct\begin{array}{lccrccl} \begin{array}{l} \mathrm{Multiply}\,3y-1\,\mathrm{by}\,-6. \\ \mathrm{Multiply}\,3y-1\,\mathrm{by 2y.} \\ \mathrm{Add like terms.} \\ \end{array} & & & \begin{array}{r} 3y-1 \\ \underset{\mathrm{\_\_\_\_\_\_\_\_}}{\,\times\,2y-6} \\ -18y+6 \\ \underset{\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}}{6{y}^{2}-2y} \\ 6{y}^{2}-20y+6 \\ \end{array} & & & \begin{array}{l} \mathrm{partial product} \\ \mathrm{partial product} \\ \mathrm{product} \\ \end{array} \\ \end{array}

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

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

To multiply binomials, use the:

  • Distributive Property
  • FOIL Method
  • Vertical Method

Remember, FOIL only works when multiplying two binomials.

24. Multiply a Trinomial by a Binomial

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, FOIL 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.

25. Example 6.45

Example 6.45

Example 6.45

Multiply using the Distributive Property: (b+3)(2b2−5b+8).(b+3)(2{b}^{2}-5b+8).

Solution

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

26. Example 6.46

Example 6.46

Example 6.46

Multiply using the Vertical Method: (b+3)(2b2−5b+8).(b+3)(2{b}^{2}-5b+8).

Solution

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

We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

To multiply a trinomial by a binomial, use the:

  • Distributive Property
  • Vertical Method

27. Section 6.3 Exercises

Section 6.3 Exercises

Section 6.3 Exercises

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

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

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

Multiply a Trinomial by a Binomial

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

In the following exercises, multiply. Use either method.

Mixed Practice

Everyday Math

Writing Exercises

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

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

28. Practice 1

  1. Multiply: 5(x+7).5(x+7).
  1. Multiply: 3(y+13).3(y+13).
  1. Multiply: x(x−7).x(x-7).
  1. Multiply: d(d−11).d(d-11).
  1. Multiply: 5x(x+4y).5x(x+4y).
  1. Multiply: 2p(6p+r).2p(6p+r).
  1. Multiply: −3y(5y2+8y−7).-3y(5{y}^{2}+8y-7).

29. Practice 2

  1. Multiply: 4x2(2x2−3x+5).4{x}^{2}(2{x}^{2}-3x+5).
  1. Multiply: 4x(3x2−5x+3).4x(3{x}^{2}-5x+3).
  1. Multiply: −6a3(3a2−2a+6).-6{a}^{3}(3{a}^{2}-2a+6).
  1. Multiply: (x+8)p.(x+8)p.
  1. Multiply: (a+4)p.(a+4)p.
  1. Multiply: (x+8)(x+9).(x+8)(x+9).
  1. Multiply: (5x+9)(4x+3).(5x+9)(4x+3).

30. Practice 3

  1. Multiply: (3b+5)(4b+6).(3b+5)(4b+6).
  1. Multiply: (a+10)(a+7).(a+10)(a+7).
  1. Multiply: (5y+2)(6y−3).(5y+2)(6y-3).
  1. Multiply: (3c+4)(5c−2).(3c+4)(5c-2).
  1. Multiply: (a+7)(a−b).(a+7)(a-b).
  1. Multiply: (x+5)(x−y).(x+5)(x-y).
  1. Multiply using the FOIL method: (x+6)(x+8).(x+6)(x+8).

31. Practice 4

  1. Multiply using the FOIL method: (y+17)(y+3).(y+17)(y+3).
  1. Multiply: (x−7)(x+5).(x-7)(x+5).
  1. Multiply: (b−3)(b+6).(b-3)(b+6).
  1. Multiply: (3x+7)(5x−2).(3x+7)(5x-2).
  1. Multiply: (4y+5)(4y−10).(4y+5)(4y-10).
  1. Multiply: (10c−d)(c−6).(10c-d)(c-6).
  1. Multiply: (7x−y)(2x−5).(7x-y)(2x-5).

32. Practice 5

  1. Multiply: (x2+6)(x−8).({x}^{2}+6)(x-8).
  1. Multiply: (y2+7)(y−9).({y}^{2}+7)(y-9).
  1. Multiply: (2ab+5)(4ab−4).(2ab+5)(4ab-4).
  1. Multiply: (2xy+3)(4xy−5).(2xy+3)(4xy-5).
  1. Multiply using the Vertical Method: (5m−7)(3m−6).(5m-7)(3m-6).
  1. Multiply using the Vertical Method: (6b−5)(7b−3).(6b-5)(7b-3).
  1. Multiply using the Distributive Property: (y−3)(y2−5y+2).(y-3)({y}^{2}-5y+2).

33. Practice 6

  1. Multiply using the Distributive Property: (x+4)(2x2−3x+5).(x+4)(2{x}^{2}-3x+5).
  1. Multiply using the Vertical Method: (y−3)(y2−5y+2).(y-3)({y}^{2}-5y+2).
  1. Multiply using the Vertical Method: (x+4)(2x2−3x+5).(x+4)(2{x}^{2}-3x+5).
  1. 4(w+10)4(w+10)
  1. −3(a+7)-3(a+7)
  1. 2(x−7)2(x-7)
  1. −3(k−4)-3(k-4)

34. Practice 7

  1. q(q+5)q(q+5)
  1. −b(b+9)-b(b+9)
  1. −x(x−10)-x(x-10)
  1. 6r(4r+s)6r(4r+s)
  1. 12x(x−10)12x(x-10)
  1. −9a(3a+5)-9a(3a+5)
  1. 3(p2+10p+25)3({p}^{2}+10p+25)

35. Practice 8

  1. −8x(x2+2x−15)-8x({x}^{2}+2x-15)
  1. 5q3(q3−2q+6)5{q}^{3}({q}^{3}-2q+6)
  1. −8y(y2+2y−15)-8y({y}^{2}+2y-15)
  1. 5q3(q2−2q+6)5{q}^{3}({q}^{2}-2q+6)
  1. −4z2(3z2+12z−1)-4{z}^{2}(3{z}^{2}+12z-1)
  1. (2m−9)m(2m-9)m
  1. (w−6)⋅8(w-6)\cdot{}8

36. Practice 9

  1. 4(x+10)4(x+10)
  1. 15(r−24)15(r-24)
  1. −3(m+11)-3(m+11)
  1. −8(z−5)-8(z-5)
  1. u(u+5)u(u+5)
  1. n(n2−3n)n({n}^{2}-3n)
  1. 6x(4x+y)6x(4x+y)

37. Practice 10

  1. 5p(11p−5q)5p(11p-5q)
  1. 3(v2+10v+25)3({v}^{2}+10v+25)
  1. 2n(4n2−4n+1)2n(4{n}^{2}-4n+1)
  1. −8y(y2+2y−15)-8y({y}^{2}+2y-15)
  1. 5q3(q2−2q+6)5{q}^{3}({q}^{2}-2q+6)
  1. −4z2(3z2+12z−1)-4{z}^{2}(3{z}^{2}+12z-1)
  1. (2y−9)y(2y-9)y

38. Practice 11

  1. (w+5)(w+7)(w+5)(w+7)
  1. (p+11)(p−4)(p+11)(p-4)
  1. (x+8)(x+3)(x+8)(x+3)
  1. (y−6)(y−2)(y-6)(y-2)
  1. (w−4)(w+7)(w-4)(w+7)
  1. (p+12)(p−5)(p+12)(p-5)
  1. (6p+5)(p+1)(6p+5)(p+1)

39. Practice 12

  1. (2t−9)(10t+1)(2t-9)(10t+1)
  1. (5x−y)(3x−6)(5x-y)(3x-6)
  1. (a+b)(2a+3b)(a+b)(2a+3b)
  1. (4z−y)(z−6)(4z-y)(z-6)
  1. (x2+3)(x+2)({x}^{2}+3)(x+2)
  1. (x2+8)(x2−5)({x}^{2}+8)({x}^{2}-5)
  1. (5ab−1)(2ab+3)(5ab-1)(2ab+3)

40. Practice 13

  1. (6pq−3)(4pq−5)(6pq-3)(4pq-5)
  1. (3rs−7)(3rs−4)(3rs-7)(3rs-4)
  1. (x+5)(x2+4x+3)(x+5)({x}^{2}+4x+3)
  1. (y+8)(4y2+y−7)(y+8)(4{y}^{2}+y-7)
  1. (w−7)(w2−9w+10)(w-7)({w}^{2}-9w+10)
  1. (3q+1)(q2−4q−5)(3q+1)({q}^{2}-4q-5)
  1. (10y−6)+(4y−7)(10y-6)+(4y-7)

41. Practice 14

  1. (x2−4x−34)−(x2+7x−6)({x}^{2}-4x-34)-({x}^{2}+7x-6)
  1. 5q(3q2−6q+11)5q(3{q}^{2}-6q+11)
  1. 8t(2t2−5t+6)8t(2{t}^{2}-5t+6)
  1. (s−7)(s+9)(s-7)(s+9)
  1. (y2−2y)(y+1)({y}^{2}-2y)(y+1)
  1. (3n−4)(n2+n−7)(3n-4)({n}^{2}+n-7)
  1. (7p+10)(7p−10)(7p+10)(7p-10)

42. Practice 15

  1. (4m2−3m−7)m2(4{m}^{2}-3m-7){m}^{2}
  1. (5a+7b)(5a+7b)(5a+7b)(5a+7b)
  1. (4y+12z)(4y−12z)(4y+12z)(4y-12z)
  1. 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 20−220-2 and 17 as 20−320-3.
  1. ⓐ Multiply (20−2)(20−3)(20-2)(20-3) by the FOIL method.
  1. ⓑ Multiply 18⋅1718\cdot{}17 without using a calculator.
  1. ⓒ Which way is easier for you? Why?
  1. Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?
  1. Multiply the following:

(m−3)(m+3)(n−10)(n+10)(p−8)(p+8)\begin{array}{l} (m-3)(m+3) \\ (n-10)(n+10) \\ (p-8)(p+8) \\ \end{array}

Explain the pattern that you see in your answers.

  1. Multiply the following:

(x−4)(x−4)(y−1)(y−1)(z−7)(z−7)\begin{array}{l} (x-4)(x-4) \\ (y-1)(y-1) \\ (z-7)(z-7) \\ \end{array}

Explain the pattern that you see in your answers.

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