Given p(x) = x4 – 13x2 – 25x – 12. A) Use long division to determine the remainder when p(x) is divided by (x2 – 4)? Show all of your work for full credit.

B) Use mathematical methods to prove that your quotient and remainder are correct. Show all of your work for full credit.

Below.

Step-by-step explanation:

p(x) = x^4 - 13x^2 – 25x – 12

Long division:

x^2 - 4)x^4 - 13x^2 – 25x – 12( x^2 - 9  < Quotient

           x^4 -  4x^2

Subtract:       -9x^2 -  25x

                     -9x^2  + 36

                                 -25x - 36 - 12

So the remainder is -25x - 48 (Answer).

B)   If the above are correct then:

(x^2 - 4) * quotient + remainder  = the original form of  p(x).

(x^2 - 4)(x^2 -9)  - 25x - 48 = the original p(x).

p(x) = x^4 - 9x^2 - 4x^2 + 36  - 25x - 48

= x^4 - 13x^2 - 25x -12 = original p(x)

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