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Before you get started, take this readiness quiz.
We have used the Distributive Property to simplify expressions like $2(x-3)$ . You multiplied both terms in the parentheses, $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3$ , by 2, to get $2x-6$ . With this chapter’s new vocabulary, you can say you were multiplying a binomial, $x-3$ , by a monomial, 2.
Multiplying a binomial by a monomial is nothing new for you! Here’s an example:
Multiply: $4\left(x+3\right).$
Distribute. | |
Simplify. |
Multiply: $y\left(y-2\right).$
Distribute. | |
Simplify. |
Multiply: $\mathrm{-2}y\left(4{y}^{2}+3y-5\right).$
Distribute. | |
Simplify. |
Multiply: $\mathrm{-3}y\left(5{y}^{2}+8y-7\right).$
$\mathrm{-15}{y}^{3}-24{y}^{2}+21y$
Multiply: $4{x}^{2}(2{x}^{2}-3x+5).$
$8{x}^{4}-24{x}^{3}+20{x}^{2}$
Multiply: $2{x}^{3}({x}^{2}-8x+1).$
Distribute. | |
Simplify. |
Multiply: $4x\left(3{x}^{2}-5x+3\right).$
$12{x}^{3}-20{x}^{2}+12x$
Multiply: $\mathrm{-6}{a}^{3}(3{a}^{2}-2a+6).$
$\mathrm{-18}{a}^{5}+12{a}^{4}-36{a}^{3}$
Multiply: $\left(x+3\right)p.$
The monomial is the second factor. | |
Distribute. | |
Simplify. |
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.
Look at [link] , where we multiplied a binomial by a monomial .
We distributed the p to get: | |
What if we have ( x + 7) instead of p ? | |
Distribute ( x + 7). | |
Distribute again. | |
Combine like terms. |
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.
Multiply: $\left(y+5\right)\left(y+8\right).$
Distribute ( y + 8). | |
Distribute again | |
Combine like terms. |
Multiply: $\left(x+8\right)\left(x+9\right).$
${x}^{2}+17x+72$
Multiply: $\left(5x+9\right)\left(4x+3\right).$
$20{x}^{2}+51x+27$
Multiply: $\left(2y+5\right)\left(3y+4\right).$
Distribute (3 y + 4). | |
Distribute again | |
Combine like terms. |
Multiply: $\left(3b+5\right)\left(4b+6\right).$
$12{b}^{2}+38b+30$
Multiply: $\left(a+10\right)\left(a+7\right).$
${a}^{2}+17a+70$
Multiply: $\left(4y+3\right)\left(2y-5\right).$
Distribute. | |
Distribute again. | |
Combine like terms. |
Multiply: $\left(5y+2\right)\left(6y-3\right).$
$30{y}^{2}-3y-6$
Multiply: $\left(3c+4\right)\left(5c-2\right).$
$15{c}^{2}+14c-8$
Multiply: $(x+2)(x-y).$
Distribute. | |
Distribute again. | |
There are no like terms to combine. |
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 [link] , 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.
Where did the first term, ${x}^{2}$ , come from?
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘ F irst, O uter, I nner, L ast’. The word FOIL is easy to remember and ensures we find all four products.
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