# Multiplication of Polynomials – Example and Practice Problems

Polynomial multiplication exercises can be solved by using the distributive property and multiplying each term of the first polynomial by each term of the second polynomial. Every time we multiply polynomials, we get a polynomial with a higher degree.

Here, we will look at a summary of the multiplication of polynomials. In addition, we will explore examples with answers of polynomial multiplication to fully understand the process used to obtain the solution.

##### ALGEBRA

Relevant for

Learning to solve polynomial multiplication problems.

See examples

##### ALGEBRA

Relevant for

Learning to solve polynomial multiplication problems.

See examples

## Summary of multiplication of polynomials

Polynomial multiplication can be a bit more complicated than polynomial addition or subtraction. We have to use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. Recall that the following is the distributive property:

The number or algebraic expression has to be distributed to each term of the polynomial. For example, we can distribute the 3 in $latex 3(x+5)$ to obtain the equivalent expression $latex 3x+15$.

The resulting polynomial is simplified by adding or subtracting like terms. Every time we multiply polynomials, we always get a polynomial with a higher degree.

Therefore, to multiply polynomials, we simply follow two steps:

Step 1: Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Step 2: We simplify by combining like terms.

## Multiplication of polynomials – Examples with answers

The following examples have their respective solution to understand the process used to arrive at the answer. It is advisable to try to solve the exercises yourself before looking at the answer.

### EXAMPLE 1

Multiply the polynomial $latex 2x+3y-5$ by $latex 2{{x}^2}$.

We multiply each term of the polynomial $latex 2x+3y-5$ by the monomial $latex 2{{x}^2}$:

⇒ $latex 2{{x}^2}(2x+3y-5)$

$$=(2{{x}^2})(2x)+(2{{x}^2})(3y)+(2{{x}^2})(-5)$$

$latex =4{{x}^3}+6{{x}^2}y-10x^2$

In this case, we do not have like terms, so we cannot simplify.

### EXAMPLE 2

Multiply the polynomial $latex 2x+4y-5z$ by $latex -4x$.

We multiply each term of the polynomial by the monomial:

⇒  $latex -4x(2x+4y-5z)$

$$=-(4x)(2x)-(4x)(4y)-(4x)(-5z)$$

$latex =-8{{x}^2}-16xy+20xz$

In this case, we don’t have like terms either, so we can’t simplify.

### EXAMPLE 3

Multiply $latex 5{{p}^3}-6pq+4{{q}^2}$ by $latex 3pq$.

We have to apply the distributive property to multiply the monomial by each of the terms of the polynomial:

⇒  $latex 3pq(5{{p}^3}-6pq+4{{q}^2})$

$$=3pq(5{{p}^3})+3pq(-6pq)+3pq(4{{q}^2})$$

$$=15{{p}^4}q-18{{p}^2}{{q}^2}+12p{{q}^3}$$

### EXAMPLE 4

Multiply the polynomials $latex (x+2)$ y $latex ({{x}^2}+5x+2)$.

Here, we have a binomial that multiplies a trinomial. We have to multiply each term of the binomial by each term of the trinomial. Therefore, we can start by separating the binomial:

⇒  $latex (x+2)({{x}^2}+5x+2)$

$latex =x({{x}^2}+5x+2)+2({{x}^2}+5x+2)$

Now, we apply the distributive property twice:

$latex ={{x}^3}+5{{x}^2}+2x+2{{x}^2}+10x+4$

In this case, we do have like terms, so we combine them to simplify to the polynomial:

$latex ={{x}^3}+7{{x}^2}+12x+4$

### EXAMPLE 5

Multiply $latex (2{{x}^2}-4)$ by $latex (-3{{x}^2}+4x-10)$.

Similar to the previous exercise, we have to separate the binomial so that each term multiplies the trinomial:

⇒  $latex (2{{x}^2}-4)(-3{{x}^2}+4x-10)$

$$=2{{x}^2}(-3{{x}^2}+4x-10) -4(-3{{x}^2}+4x-10))$$

Now, we apply the distributive property so that each term of the binomial is multiplied by each term of the trinomial:

$$=(2{{x}^2})(-3{{x}^2})+(2{{x}^2})(4x)+(2{{x}^2})(-10) -4(-3{{x}^2})-4(4x)-4(-10)$$

$$=-6{{x}^4}+8{{x}^3}-20{{x}^2}+12{{x}^2}-16x+40$$

We can combine like terms to simplify:

$$=-6{{x}^4}+8{{x}^3}-8{{x}^2}-16x+40$$

### EXAMPLE 6

Multiply the polynomials $latex (2x+5y)$ and $latex (3{{x}^2}+5xy+4{{y}^2})$.

We separate the binomial to multiply it by the trinomial:

⇒  $latex (2x+5y)(3{{x}^2}+5xy+10{{y}^2})$

$$=2x(3{{x}^2}+5xy+10{{y}^2}) +5y(3{{x}^2}+5xy+10{{y}^2})$$

Now, we multiply each term using the distributive property:

$$=(2x)(3{{x}^2})+(2x)(5xy)+(x)(10{{y}^2}) +(5y)(3{{x}^2})+(5y)(5xy)+(5y)(10{{y}^2})$$

$$=6{{x}^2}+10{{x}^2}y+10x{{y}^2} +15{{x}^2}y+25x{{y}^2}+50{{y}^3}$$

We can combine like terms to simplify:

$$=6{{x}^2}+25{{x}^2}y+35x{{y}^2}+50{{y}^3}$$

## Multiplication of polynomials – Practice problems

Put what you have learned into practice with the following polynomial multiplication problems. Solve the exercises, choose your answer and check it to verify that you selected the correct one.