## Multiplying Two Polynomials

### Let's Review

First, let's review the definition of a polynomial: It's an expression of variables and constants combined using addition, subtraction, multiplication, and division. For example, the following are simple polynomials:

- \(x+2\)
- \(x^2-3x-9\)
- \(x^5-x^3+7x+30\)

So, what if we want to multiply two polynomials together? Say we want to do the following:

$$ (x^2-2x+1)*(3x^3+4x^2+5x)=? $$We can use the FOIL method to multiply polynomials, but that is often demonstrated only for simpler, two-term polynomials. To multiply out a larger polynomial (often called expanding), we need a more generalized understanding of how to multiply polynomials. We will use the distributive property to distribute the terms from one of the polynomials being multiplied to the other. In other words, we must take every term from the first polynomial and multiply it by each term in the other polynomial. So, as you can imagine, we have a large number of resulting terms! Let's work our way step-by-step through the first polynomial:

Step 1: Multiply \(x^2\) by every term in the second expression:

$$ x^2(3x^3+4x^2+5x)=3x^5+4x^4+5x^3 $$Step 2: Multiply \(-2x\) by every term in the second expression.

$$ -2x(3x^3+4x^2+5x)=-6x^4-8x^3-10x^2 $$Step 3: Multiply 1 by every term in the second expression.

$$ 1(3x^3+4x^2+5x)=3x^3+4x^2+5x $$Step 4: Now we take those nine terms (everything on the right side of those above equations) and add 'em all up and combine like terms:

$$ 3x^5+4x^4+5x^3-6x^4-8x^3-10x^2+3x^3+4x^2+5x= $$Final answer:

$$ 3x^5-2x^4-6x^2+5x $$This is how you multiply together any two polynomials. Just remember to multiply everything in one polynomial by every term in the other polynomial and then add up what you get. That's really just a simple way of saying "distribute the terms of one polynomial to the other."

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