There are several different methods you can use to multiply polynomials.
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
- Identify the terms of each polynomial.
- Multiply each term in the first polynomial by every term in the second polynomial (or vice versa).
- Combine like terms, if needed.
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.
Then I’ll combine like terms to simplify my answer.
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+2 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:
Then you can combine the result of making 2 copies and x copies to simplify the answer for (x+2)(3x+4).
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.