How to Multiply Polynomials

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Summary

There are several different methods you can use to multiply polynomials. 

My favorite method is the multiplication algorithm. A lot of my students prefer the box method. These methods can be used to multiply any kind of polynomial. 

If you're multiplying monomials by polynomials, you can use the distributive property. And if you're multiplying binomials by binomials, you can use the FOIL method.

The big idea behind each of these methods is that every term in the first polynomial has to be multiplied by the every term in the second polynomial. 

How to Multiply Polynomials

  1. Identify the terms of each polynomial. 
  2. Multiply each term in the first polynomial by every term in the second polynomial (or vice versa).
  3. Combine like terms, if needed. 

If there are any negative coefficients in the polynomials, be sure to follow the rules for multiplying negative numbers. If you don't know how to multiply variables, check out this page.

Example

Multiply \((x^2-3x+9)(5x-2)\).

I'll start by identifying the terms of the polynomials.

First Polynomial: \(x^2\), \(-3x\), and \(9\)

Second Polynomial: \(5x\) and \(-2\)

Next, I'll multiply each term in the first polynomial by each term in the second polynomial. 

\(x^2(5x-2)=5x^3-2x^2\)

\(-3x(5x-2)=-15x^2+6x\)

\(9(5x-2)=45x-18\)

Then I'll combine like terms to simplify my answer.

\(5x^3-2x^2-15x^2+6x+45x-18\)

\(5x^3-17x^2+51x-18\)

Answer:

\((x^2-3x+9)(5x-2)=5x^3-17x^2+51x-18\)

Why It Works

The process for multiplying polynomials is basically an extended version of the distributive property.

The distributive property says that 2(3x+4)=6x+8 because if you double a group of 3 x's and 4 units the result is equivalent to 6 x's and 8 units. 

When you multiply polynomials like (x+2)(3x+4), you are basically making x+copies of (3x+4).

That means that you will make 2 copies of the group:

\(2(3x+4) = 6x+8\)

And you will also make x copies of the group:

\(x(3x+4)=3x^2+4x\)

Then you can combine the result of making 2 copies and x copies to simplify the answer for (x+2)(3x+4).

\((x+2)(3x+4)=3x^2+10x+8\)

So, the process for multiplying polynomials is essentially the same as the distributive property. The only difference is that you need to distribute multiple terms instead of just one term.

This process is the same even if you have longer polynomials. Just make sure to multiply EACH of the terms in the first polynomial by ALL of the terms in the second polynomial. 

Resources

Free Online Practice Problems

Khan Academy - Multiply Binomials by Polynomials

Khan Academy - Multiply Binomials

Khan Academy - Multiply Monomials by Polynomials


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