Polynomials

When polynomials are multiplied, each term of one expression is multiplied by every term of the other expression. Then the terms in the product are added.

Choose one polynomial, preferably the longest, and then:

• multiply it by the first term of the other polynomial
• then multiply it by the second term of the other polynomial then multiply it by the third term of the other polynomial (if any) etc ...
• lastly, add up the products.

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Example

Multiply the following polynomials:

$$(a+b+c)\cdot (a+b)$$

We first multiply our first polynomial by the first term of the second polynomial then we multiply it by the second term of the other polynomial:

$$(a+b+c)\cdot (a+b)=$$

$$=a\cdot a+b\cdot a+c\cdot a+a\cdot b+b\cdot b+c\cdot b$$

Lastly we simplify and add up the products

$$a\cdot a+b\cdot a+c\cdot a+a\cdot b+b\cdot b+c\cdot b=$$ $$a^{2}+2ab+ac+b^{2}+cb$$

When we are dividing a polynomial by something more complicated than just a simple monomial, then we will need to use a different method for the simplification. This method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables.

We will show the procedure in an example.

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Example

Divide x²-2x-8 by x+2:

First, we set up the division, writing x+2 to the left and x²-2x-8 to the right. Our result comes out on top of the polynomials. In order to see exactly how this is done watch our video lesson below.

$$x+2{\overline{\smash{\big)}\,x^2-2x-8\phantom{)}}}$$

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Video lesson

Divide x²-2x-8 by x+2 (same as the example above).