Question

Directions: Find the quotient and the remainder when the first polynomial is divided by the second.
#9. 3x^4 – 2x^3 + 5x^2 + x + 1;x^2 + 2x

Answer 1

Do exactly as you would do in a long division of numbers:

3x^4 – 2x^3 + 5x^2 + x + 1;x^2 + 2x

I will do it below, but the coefficients will not be lined up because I do not know how to insert spaces that don't get merged at this forum.
Divide x^2+2x into

3x^4 – 2x^3 + 5x^2 + x + 1

First divide 3x^4 by x^2 to get <b>3x^2</b>.

Then multiply (x^2+2x) by 3x^2 to get
3x^4+6x^3

Subtract 3x^4+6x^3 from 3x^4 – 2x^3 + 5x^2 + x + 1 to get
– 8x^3 + 5x^2 + x + 1

Repeat the same as above:
Divide -8x^3 by x^2 to get <b>-8x</b>

Multiply x^2+2x by -8x to get -8x^3 -16x^2.

Subtract -8x^3 -16x^2 from – 8x^3 + 5x^2 + x + 1 to get 21x^2 +x + 1.

The same process will repeat itself to get the final answer by collecting all the above quotient terms:
3x^2-8x+21 with a remainder of -41x+1