Previously, you learned to use the Distributive Property to simplify expressions such as [latex]2\left(x - 3\right)[/latex]. You multiplied both terms in the parentheses, [latex]x\text{ and }3[/latex], by [latex]2[/latex], to get [latex]2x - 6[/latex]. With this chapter’s new vocabulary, you can say you were multiplying a binomial, [latex]x - 3[/latex], by a monomial, [latex]2[/latex]. Multiplying a binomial by a monomial is nothing new for you!

example

Multiply: [latex]3\left(x+7\right)[/latex]

Solution

	[latex]3\left(x+7\right)[/latex]
Distribute.
	[latex]3\cdot x+3\cdot 7[/latex]
Simplify.	[latex]3x+21[/latex]

example

Multiply: [latex]x\left(x - 8\right)[/latex]

Show Solution

Solution

	[latex]x(x-8)[/latex]
Distribute.
	[latex]x^2-8x[/latex]
Simplify.	[latex]x^2-8x[/latex]

example

Multiply: [latex]10x\left(4x+y\right)[/latex]

Show Solution

Solution

	[latex]10x(4x+y)[/latex]
Distribute.
	[latex]10x\cdot{4x}+10x\cdot{y}[/latex]
Simplify.	[latex]40x^2+10xy[/latex]

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

example

Multiply: [latex]-2x\left(5{x}^{2}+7x - 3\right)[/latex]

Show Solution

Solution

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

example

Multiply: [latex]4{y}^{3}\left({y}^{2}-8y+1\right)[/latex]

Show Solution

Solution

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

Now we will have the monomial as the second factor.

example

Multiply: [latex]\left(x+3\right)p[/latex]

Show Solution

Solution

	[latex]\left(x+3\right)p[/latex]
Distribute.
	[latex]x\cdot p+3\cdot p[/latex]
Simplify.	[latex]xp+3p[/latex]

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