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I have had a problem with this question for a while now. The reason to be is that I do not understand what I have to do! The thing is we have been specifically told not to use long division.

Here is the question:

The polynomial $f(x)$ is given by $f(x) =4x^3 +ax^2 + bx+2$. $(4x+1)$ is a factor of $f(x)$.When $f(x)$ is divided by $(x-1)$ there is a remainder of $20$. Find $a$ and $b$.

Please help; I'm so stuck.

Carl Schildkraut
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Jack Black
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  • Well first I did the following: f(-1/4) and then put that value into the equation and got like (a-4b+31)/16 and for the second equation I got a+4+b+2=20 so a+b= 14. I solved them simultaneously,but I know for a fact the result is incorrect..so I'm still stuck – Monica 9 mins ago delete – Jack Black Sep 25 '16 at 18:24

1 Answers1

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Hint 1: If $(4x+1)|(4x^3+ax^2+bx+2)$, you can write

$$4x^3+ax^2+bx+2 = (4x+1)P(x)$$

for some polynomial $P(x)$. What can you say about its coefficients?

Hint 2: If $f(x)$ leaves a remainder of $20$ when divided by $x-1$, you can write

$$f(x) = (x-1)Q(x)+20$$

for some polynomial $Q(x)$. What can you say about its coefficients?

Carl Schildkraut
  • 32,931