Consider the following division of polynomials. 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.

. Multiplying the denominator by gives

Subtracting this from the numerator gives a remainder of

. Multiplying the denominator by gives

and subtracting this from the previous remainder gives a new remainder of

This last remainder is exactly the same as the denominator, so divides through it exactly and leaves us with 1.

What we showed here is that

and this last expression is the quotient.

To verify this solution, we can simply multiply this by the original denominator:

which matches the original numerator.

Step-by-step explanation:

A) Long division :

x²+2x+8) x⁴+ x³+ 7x²-6x+8 ( x²-x+1

               x⁴+2x³+8x²

                        -x³ -x² -6x

                        -x³ -x² -8x

                              x² +2x + 8

                              x² +2x + 8

                                   NIL      

B)  By multiplying (x² + 2x + 8) and quotient (x²-x+1) We should get the expression (x⁴+x³+7x²-6x+8) as of numerator.

(x²+2x+8) × (x²-x+1)

= x²(x²-x+1) + 2x(x²-x+1)+8(x²-x+1)

= x⁴-x³+x²+2x³-2x²+2x+8x²-8x+8

= (x⁴+x³+7x²-6x+8)

Hence proved.

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