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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.
Multiply using the Distributive Property: $(b+3)(2{b}^{2}-5b+8).$
Distribute. | |
Multiply. | |
Combine like terms. |
Multiply using the Distributive Property: $(y-3)({y}^{2}-5y+2).$
${y}^{3}-8{y}^{2}+17y-6$
Multiply using the Distributive Property: $(x+4)(2{x}^{2}-3x+5).$
$2{x}^{3}+5{x}^{2}-7x+20$
Now let’s do this same multiplication using the Vertical Method.
Multiply using the Vertical Method: $(b+3)(2{b}^{2}-5b+8).$
It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
Multiply (2 b ^{2} − 5 b + 8) by 3. | |
Multiply (2 b ^{2} − 5 b + 8) by b. | |
Add like terms. |
Multiply using the Vertical Method: $(y-3)({y}^{2}-5y+2).$
${y}^{3}-8{y}^{2}+17y-6$
Multiply using the Vertical Method: $(x+4)(2{x}^{2}-3x+5).$
$2{x}^{3}+5{x}^{2}-7x+20$
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:
Access these online resources for additional instruction and practice with multiplying polynomials:
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
$\mathrm{-5}(p+9)$
$\mathrm{-8}(j-5)$
$\text{\u2212}y(y+3)$
$\text{\u2212}p(p-15)$
$5c(9c+d)$
$9m(m-11)$
$\mathrm{-4}p(2p+7)$
$6({y}^{2}+8y+16)$
$\mathrm{-8}x({x}^{2}+2x-15)$
$\mathrm{-8}{x}^{3}-16{x}^{2}+120x$
$\mathrm{-5}t({t}^{2}+3t-18)$
$5{q}^{3}\left({q}^{3}-2q+6\right)$
$5{q}^{6}-10{q}^{4}+30{q}^{3}$
$4{x}^{3}\left({x}^{4}-3x+7\right)$
$\mathrm{-8}y({y}^{2}+2y-15)$
$\mathrm{-8}{y}^{3}-16{y}^{2}+120y$
$\mathrm{-5}m({m}^{2}+3m-18)$
$9{r}^{3}({r}^{2}-3r+5)$
$\mathrm{-4}{z}^{2}(3{z}^{2}+12z-1)$
$\mathrm{-12}{z}^{4}-48{z}^{3}+4{z}^{2}$
$\mathrm{-3}{x}^{2}(7{x}^{2}+10x-1)$
$(8j-1)j$
$(k-4)\xb75$
$12(v-30)$
$\mathrm{-4}(p+15)$
$\mathrm{-3}(x-9)$
$s({s}^{2}-6s)$
$5a(9a+b)$
$12u(3u-4v)$
$6({x}^{2}+8x+16)$
$3r(2{r}^{2}-6r+2)$
$\mathrm{-8}y({y}^{2}+2y-15)$
$\mathrm{-8}{y}^{3}-16{y}^{2}+120y$
$\mathrm{-5}m({m}^{2}+3m-18)$
$9{r}^{3}({r}^{2}-3r+5)$
$\mathrm{-4}{z}^{2}(3{z}^{2}+12z-1)$
$\mathrm{-12}{z}^{4}-48{z}^{3}+4{z}^{2}$
$\mathrm{-3}{x}^{2}(7{x}^{2}+10x-1)$
$(8b-1)b$
Multiply a Binomial by a Binomial
In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.
$(y+9)(y+3)$
$(q+4)(q-8)$
In the following exercises, multiply the binomials. Use any method.
$(y+7)(y+4)$
$(x-7)(x-2)$
$(q-5)(q+8)$
$(m+11)(m-4)$
$(7m+1)(m+3)$
$(3r-8)(11r+1)$
$\left(10a-b\right)\left(3a-4\right)$
$(r+s)(3r+2s)$
$(5x-y)(x-4)$
$({y}^{2}-4)(y+3)$
$({y}^{2}-7)({y}^{2}-4)$
$(2xy+3)(3xy+2)$
$(3rs-7)(3rs-4)$
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.
$(u+4)({u}^{2}+3u+2)$
$(a+10)(3{a}^{2}+a-5)$
In the following exercises, multiply. Use either method.
$(p-4)({p}^{2}-6p+9)$
$(6r+1)({r}^{2}-7r-9)$
Mixed Practice
$\left(15p-4\right)+\left(3p-5\right)$
$\left({x}^{2}-4x-34\right)-\left({x}^{2}+7x-6\right)$
$\mathrm{-11}x-28$
$\left({j}^{2}-8j-27\right)-\left({j}^{2}+2j-12\right)$
$8t(2{t}^{2}-5t+6)$
$\left(x-5\right)\left(x+13\right)$
$\left({y}^{2}-2y\right)\left(y+1\right)$
${y}^{3}-{y}^{2}-2y$
$\left({a}^{2}-3a\right)\left(4a+5\right)$
$\left(3n-4\right)\left({n}^{2}+n-7\right)$
$3{n}^{3}-{n}^{2}-25n+28$
$\left(6k-1\right)\left({k}^{2}+2k-4\right)$
$\left(3y+8\right)\left(3y-8\right)$
$(15{c}^{2}-4c+5){c}^{4}$
$\left(5a+7b\right)\left(5a+7b\right)$
$25{a}^{2}+70ab+49{b}^{2}$
$\left(3x-11y\right)\left(3x-11y\right)$
$\left(4y+12z\right)\left(4y-12z\right)$
$16{y}^{2}-144{z}^{2}$
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 $10+5$ .
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 $20-3$ .
ⓐ 306 ⓑ 306 ⓒ Answers will vary.
Which method do you prefer to use when multiplying two binomials: the Distributive Property, the FOIL method, or the Vertical Method? Why?
Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?
Answers will vary.
Multiply the following:
$\begin{array}{c}(x+2)(x-2)\hfill \\ (y+7)(y-7)\hfill \\ (w+5)(w-5)\hfill \end{array}$
Explain the pattern that you see in your answers.
Multiply the following:
$\begin{array}{c}(m-3)(m+3)\hfill \\ (n-10)(n+10)\hfill \\ (p-8)(p+8)\hfill \end{array}$
Explain the pattern that you see in your answers.
Answers may vary.
Multiply the following:
$\begin{array}{c}(p+3)(p+3)\hfill \\ (q+6)(q+6)\hfill \\ (r+1)(r+1)\hfill \end{array}$
Explain the pattern that you see in your answers.
Multiply the following:
$\begin{array}{c}(x-4)(x-4)\hfill \\ (y-1)(y-1)\hfill \\ (z-7)(z-7)\hfill \end{array}$
Explain the pattern that you see in your answers.
Answers may vary.
ⓐ 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?
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