SOLUTION: given that the remainder when f(x) is divided by (x-1) is equal to the remainder when f(x) is divided by (2x+1). f(x)= px^3 + 6x^2 + 12x + q find the value of p thank you!!!!

Algebra ->  Functions -> SOLUTION: given that the remainder when f(x) is divided by (x-1) is equal to the remainder when f(x) is divided by (2x+1). f(x)= px^3 + 6x^2 + 12x + q find the value of p thank you!!!!      Log On


   



Question 265051: given that the remainder when f(x) is divided by (x-1) is equal to the remainder when f(x) is divided by (2x+1). f(x)= px^3 + 6x^2 + 12x + q
find the value of p
thank you!!!!

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
given that the remainder when f(x) is divided by (x-1) is equal to the remainder when f(x) is divided by (2x+1). f(x)= px^3 + 6x^2 + 12x + q
find the value of p


The remainder theorem states that the remainder when a polynomial f%28x%29 
is divided by k%28x+-+a%29 is f%28a%29.

Therefore the remainder when f%28x%29 is divided by x-1 is f%281%29, which is

%22f%281%29%22+=+p%2A%281%29%5E3+%2B+6%2A%281%29%5E2+%2B+12%2A%281%29+%2B+q 
%22f%281%29%22+=+p+%2B+6+%2B+12+%2B+q
%22f%281%29%22+=+p+%2B+18+%2B+q

Also the remainder when f%28x%29 is divided by 2x%2B1, or 2%28x%2B1%2F2%29
is f%28-1%2F2%29, which is

f%28-1%2F2%29+=+p%2A%28-1%2F2%29%5E3+%2B+6%2A%28-1%2F2%29%5E2+%2B+12%2A%28-1%2F2%29+%2B+q 
f%28-1%2F2%29+=+p%2A%28-1%2F8%29+%2B+6%2A%281%2F4%29+-+6+%2B+q
f%28-1%2F2%29+=+-p%2F8+%2B+3%2F2+-+6+%2B+q

We are told that these are equal:

p+%2B+18+%2B+q+=+-p%2F8+%2B+3%2F2+-+6+%2B+q

8p+%2B+144+%2B+8q+=+-p+%2B+12+-+48+%2B+8q

8p+%2B+144+%2B+8q+=+-p+=+-36+%2B+8q

9p=-180

p+=+-20

Edwin