# Long Division with Polynomials

Hi everyone!

I am VERY confused on how to do long division with polynomials!!
Here is an example:

3x^3- 4x^2+ 2x +1, x+2

Any help would be much appreciated!
Thank you in advance

1 person

#### toesockshoe

$$\displaystyle \frac{3x^2-4x^2+2x+1}{x+2}$$.
Lets look at numeric division first. I'll ask you how you divide 906 by 3 by hand. Here is the usual way: You would look at 906 digit by digit. You would ask yourself, "What times 3 is 9 (the first digit in 906). 3 times 3 is nine. You know that 3 would be the first digit of your quotient. Then you would continue and say 3*0 = 0, and 3*1 =3. Thus your quotient is 301

The trick in polynomial division is to look at the dividend $$\displaystyle (3x^2-4x^2+2x+1)$$ one term at a time. So first you ask yourself what can you multiply by x+2 to get a factor of 3x^2. Well 3x times x is 3x^2. This matches the first term of your dividend. so $$\displaystyle 3x^2 * (x+2) = 3x^3 + 2x^2$$. Now you subtract $$\displaystyle 3x^3+2x^2$$ from $$\displaystyle 3x^3 -4x^2 + 2x + 1$$ You would get $$\displaystyle -6x^2+2x+1$$ Now you see what times x+2 equals the first term of $$\displaystyle -6x^2+2x+1$$. You can see that $$\displaystyle -6(x+2) = -6x^2 - 6x$$. Good. Your first terms match. Execute the following subtraction: $$\displaystyle (-6x^2+2x+1) - ( -6x^2 - 6x)$$ to get $$\displaystyle 8x + 1$$.... Continue doing this procedure until you can't multiply (x+2) by $$\displaystyle Cx^n$$ where C is a real number, and n is a positive real number. The remaining term would be your remainder.

1 person

#### romsek

MHF Helper
what toesockshoe said in picture form

1 person