SOLUTION: f(x) is a polynomial of degree 4 with f(x) + 1 divisible by (x + 1)^2 and f(x) itself divisible by x^3. What is the value of f(−4)?

Algebra ->  Finance -> SOLUTION: f(x) is a polynomial of degree 4 with f(x) + 1 divisible by (x + 1)^2 and f(x) itself divisible by x^3. What is the value of f(−4)?      Log On


   

Question 1162880: f(x) is a polynomial of degree 4 with f(x) + 1 divisible by (x + 1)^2 and f(x) itself divisible by x^3. What is the value of f(−4)?
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


f(x) is a polynomial of degree 4 which is divisible by x^3. So

f%28x%29+=+ax%5E4%2Bbx%5E3

f(x)+1 is divisible by (x+1)^2 = x^2+2X+1.

Perform the polynomial long division....

                              ax^2 + (b-2a)x + (3a-2b)
            ----------------------------------------------
   x^2+2x+1 ) ax^4 +    bx^3 +       0x^2   +   0x   +   1
              ax^4 + (2a)x^3 +       ax^2
              -------------------------------
                   (b-2a)x^3 -       ax^2 +      0x
                   (b-2a)x^3 + (2b-4a)x^2 + (b-2a)x
                   ---------------------------------------
                               (3a-2b)x^2 +  (2a-b)x +  1
                               (3a-2b)x^2 + (6a-4b)x + (3a-2b)
                               ---------------------------------
                                            (3b-4a)x + (1-3a+2b)

The remainder has to be zero, so

3b-4a+=+0
1-3a%2B2b+=+0

Solve the pair of equations (I leave that much to you) to find a=3 and b=4.

So the function is

f%28x%29+=+3x%5E4%2B4x%5E3

ANSWER:
f%28-4%29+=+3%28-4%29%5E4%2B4%28-4%29%5E3+=+3%28256%29%2B4%28-64%29+=+768-256+=+512