<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

Multiplication of Monomials by Polynomials

Distribute single terms by multiplying them by other terms

Atoms Practice
Estimated8 minsto complete
%
Progress
Practice Multiplication of Monomials by Polynomials
Practice
Progress
Estimated8 minsto complete
%
Practice Now
Distributive Property

Learning Goal

I am learning to multiply a monomial and a polynomial using the distributive property.

Warm Up

Recall the formula for the area of a rectangle Area = Length x Width.  

License: CC BY-NC 3.0

[Figure1]

How can we write a simplified expression for the area of this rectangle?

 

Action

Just as we can add and subtract polynomials, we can also multiply them.

Multiplying a Polynomial by a Monomial

When multiplying polynomials, we must remember the exponent rules. Especially important is the product rule: xnxm=xn+m.

If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any number and we apply the product rule on each variable separately.

Example A

Multiply the following monomials.

a) (2x2)(5x3)

b) (3y4)(2y2)

 

Solution

a) (2x2)(5x3)=(25)(x2x3)=10x2+3=10x5

b) (3y4)(2y2)=(32)(y4y2)=6y4+2=6y6

 

To multiply a polynomial by a monomial, we have to use the Distributive PropertyThis property says that we need to multiply each term inside the bracket by the term outside the bracket.  a(b+c)=ab+ac.

Example B

Multiply:

a) 3(x2+3x5)

b) 4x(3x27)

c) 7y(4y22y+1)

Solution

a) 3(x2+3x5)=3(x2)+3(3x)3(5)=3x2+9x15

b) 4x(3x27)=(4x)(3x2)+(4x)(7)=12x328x

c)

7y(4y22y+1)=(7y)(4y2)+(7y)(2y)+(7y)(1)=28y3+14y27y

 

Notice that when we use the Distributive Property, the problem becomes a matter of just multiplying monomials by monomials and adding all the separate parts together.

Example C

Multiply 2a2b4(3ab2+7a3b9a+3).

Solution:

Multiply the monomial by each term inside the parenthesis:

2a2b4(3ab2+7a3b9a+3)=(2a2b4)(3ab2)+(2a2b4)(7a3b)+(2a2b4)(9a)+(2a2b4)(3)=6a3b614a5b5+18a5b46a2b4

 

We could also use Algebra Tiles!

 

Lets take a look back at our warm up question. How can we use the distributive property to write a simplified expression for the area of this rectangle?

License: CC BY-NC 3.0

[Figure2]

Solution:

License: CC BY-NC 3.0

[Figure3]

 

 

Consolidation

Check out this video summarising distributive property 

 

Word Wall

Distributive Property

A method used to multiply or expand a monomial into a polynomial. When using the distributive property multiply each term inside the bracket by the term outside the bracket.  a(b+c)=ab+ac.

 

Practice

Multiply the following monomials.

  1. (2x)(7x)
  2. (10x)(3xy)
  3. (4mn)(0.5nm2)
  4. (5a2b)(12a3b3)
  5. (3xy2z2)(15x2yz3)

Multiply and simplify.

  1. 17(8x10)
  2. 2x(4x5)
  3. 9x3(3x22x+7)
  4. 3x(2y2+y5)
  5. 10q(3q2r+5r)

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

Explore More

Sign in to explore more, including practice questions and solutions for Multiplication of Monomials by Polynomials.
Please wait...
Please wait...