Previously, you learned to use the Distributive Property to simplify expressions such as $2\left(x - 3\right)$. You multiplied both terms in the parentheses, $x\text{ and }3$, 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!

example

Multiply: $3\left(x+7\right)$

Solution

	$3\left(x+7\right)$
Distribute.
	$3\cdot x+3\cdot 7$
Simplify.	$3x+21$

example

Multiply: $x\left(x - 8\right)$

Show Solution

Solution

	$x(x-8)$
Distribute.
	$x^2-8x$
Simplify.	$x^2-8x$

example

Multiply: $10x\left(4x+y\right)$

Show Solution

Solution

	$10x(4x+y)$
Distribute.
	$10x\cdot{4x}+10x\cdot{y}$
Simplify.	$40x^2+10xy$

Multiplying a monomial by a trinomial works in much the same way.

example

Multiply: $-2x\left(5{x}^{2}+7x - 3\right)$

Show Solution

Solution

	$-2x\left(5{x}^{2}+7x - 3\right)$
Distribute.
	$-2x\cdot 5{x}^{2}+\left(-2x\right)\cdot 7x-\left(-2x\right)\cdot 3$
Simplify.	$-10{x}^{3}-14{x}^{2}+6x$

example

Multiply: $4{y}^{3}\left({y}^{2}-8y+1\right)$

Show Solution

Solution

	$4{y}^{3}\left({y}^{2}-8y+1\right)$
Distribute.
	$4{y}^{3}\cdot {y}^{2}-4{y}^{3}\cdot 8y+4{y}^{3}\cdot 1$
Simplify.	$4{y}^{5}-32{y}^{4}+4{y}^{3}$

Now we will have the monomial as the second factor.

example

Multiply: $\left(x+3\right)p$

Show Solution

Solution

	$\left(x+3\right)p$
Distribute.
	$x\cdot p+3\cdot p$
Simplify.	$xp+3p$

In the following video we show more examples of how to multiply monomials with other polynomials.