Consider the following division of polynomials.

x4+x3+7x2-6x+8/x2+2x+8

A) Use long division to determine the quotient of the polynomials. Show all of your work for full credit.

B) Use mathematical methods to prove your answer. Show all of your work for full credit.

(A)

(x⁴ + x³ + 7x² - 6x + 8) : (x² + 2x + 8)

x² - x + 1

x² + 2x + 8) x⁴ + x³ + 7x² - 6x + 8

- x⁴ + 2x³ + 8x²

-x³ - x² - 6x

- - x³ - 2x² - 8x

x² + 2x + 8

- x² + 2x + 8

0

Answer: x² - x + 1

(B) Distribution Method

(x² - x + 1) (x² + 2x + 8)

= x² (x² + 2x + 8) - x (x² + 2x + 8) + 1 (x² + 2x + 8)

= x⁴ + 2x³ + 8x² - x³ - 2x² - 8x + x² + 2x + 8

= x⁴ + (2x³ - x³) + (8x² - 2x² + x²) + (-8x + 2x) + 8

= x⁴ + x³ + 7x² - 6x + 8