8.4.2: The Distributive Property

Learning Outcomes

  • Multiply polynomials using the distributive property.

Key words

  • Distributive property: [latex]a(b+c)=ab+ac[/latex] for all terms [latex]a,b,c[/latex]

Multiplying Monomials

In the last section, we practiced multiplying monomials together. In this section we will show examples of how to multiply more than just monomials.  We will multiply monomials with binomials and trinomials. We will also learn some techniques for multiplying two binomials together.

Example

Multiply. [latex]-9x^{3}\cdot 3x^{2}[/latex]

Solution

Rearrange the factors.

[latex]-9\cdot3\cdot x^{3}\cdot x^{2}[/latex]

Multiply constants. Remember that a positive number times a negative number yields a negative number.

[latex]-27\cdot x^{3}\cdot x^{2}[/latex]

Multiply variable terms. Remember to add the exponents when multiplying exponents with the same base.

[latex]\begin{array}{l}-27\cdot x^{3+2}\\-27\cdot x^{5}\end{array}[/latex]

Answer

[latex]-9x^{3}\cdot 3x^{2}=-27x^{5}[/latex]

When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents.

The Distributive Property

Previously, we used the Distributive Property to simplify expressions such as [latex]2\left(x - 3\right)[/latex]. We 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, we can say we 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!  The distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied onto each term in the binomial.

example

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

Solution

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

try it

 

example

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

Solution

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

try it

 

example

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

Solution

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

try it

 

In the next example, we multiply a second degree monomial with a binomial.  Note the use of exponent rules.

Example

Simplify. [latex]5x^2\left(4x^{2}+3x\right)[/latex]

Solution

Distribute the monomial to each term of the binomial. Multiply coefficients and variables separately.

[latex]\begin{array}{c}5x^2\left(4x^{2}+3x\right)\\5x^2\left(4x^{2}\right)+5x^2\left(3x\right)\\\text{ }\\=20x^{2+2}+15x^{2+1}\\\text{ }\\=20x^{4}+15x^{3}\end{array}[/latex]

Answer

[latex]5x^2\left(4x^{2}+3x\right)=20x^{4}+15x^{3}[/latex]

Try It

Simplify:

  1. [latex]-3x^2\left (5x^2-8x\right )[/latex]
  2. [latex]2x^4\left (9x^3-7x^2\right )[/latex]

 

Now let’s add another layer by multiplying a monomial by a trinomial. Multiplying a monomial by a trinomial works in much the same way as multiplying a monomial by a binomial.  Consider the expression [latex]2x\left(2x^{2}+5x+10\right)[/latex].

This expression can be modeled with a sketch like the one below.

2x times 2x squared equals 4x cubed. 2x times 5x equals 10x squared. 2x times 10 equals 20x.

The only difference between this example and the previous one is there is one more term to distribute the monomial to.

[latex]\begin{equation}\begin{aligned}&\;\;\;\;\,2x\left(2x^{2}+5x+10\right)\\&=2x\left(2x^{2}\right)+2x\left(5x\right)+2x\left(10\right)\\&=4x^{3}+10x^{2}+20x\end{aligned}\end{equation}[/latex]

We always need to pay attention to negative signs when we are multiplying. Watch what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.

Example

Simplify. [latex]-7x\left(2x^{2}-5x+1\right)[/latex]

Solution

[latex]-7x\left(2x^{2}-5x+1\right)[/latex]

Distribute the monomial to each term in the trinomial.

[latex]-7x\left(2x^{2}\right)-7x\left(-5x\right)-7x\left(1\right)[/latex]

 

Multiply.

[latex]\begin{array}{c}-14x^{1+2}+35x^{1+1}-7x\\\text{ }\\-14x^{3}+35x^{2}-7x\end{array}[/latex]

Rewrite addition of terms with negative coefficients as subtraction.

Answer

[latex]-7x\left(2x^{2}-5x+1\right)=-14x^{3}+35x^{2}-7x[/latex]

example

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

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]

try it

 

example

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

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]

try it

 

In the next example, the monomial is the second factor.

example

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

Solution

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

try it

 

The following video shows more examples of how to multiply monomials with other polynomials.