# Thread: [SOLVED] How to solve this question? Many thanks!

1. ## [SOLVED] How to solve this question? Many thanks!

Given that $\displaystyle \text{P(x)}\,\equiv{x^4 + x^3 + ax^2 + bx + c}$ where a, b, c are constants. If P(x) $\displaystyle \equiv{(x - 1)^2 Q(x)}$, Q(x) is a polynomial with integer coefficient and P(0) = 2, find the values of a, b, and c.

The answer they gave is a = -3, b = -1 and c = 2 not a=-5, b=3

2. $\displaystyle P(0)=2$
$\displaystyle c=2$

$\displaystyle P(1)=0$ (Why?)
$\displaystyle 1+1+a+b+c=0$
$\displaystyle a+b+4=0$
$\displaystyle a+b=-4$----(1)

$\displaystyle P'(1)=0$ (Because x=1 is a repeated root)
$\displaystyle 4+3+2a+b=0$
$\displaystyle 7+2a+b=0$
$\displaystyle 2a+b=-7$-----(2)

Solving (1) and (2),
a=-3
b=-1

a=-3, b=-1, c=2

3. Hey, what does P' means and how do you do it? Why do you put the power of 4, 3 and 2 equals to 0? thanks!

4. Originally Posted by mark1950
Hey, what does P' means and how do you do it? Why do you put the power of 4, 3 and 2 equals to 0? thanks!
$\displaystyle P'(x)$ is the differentiated polynomial of $\displaystyle P(x)$

$\displaystyle P'(x)=4x^3+3x^2+2ax+b$

$\displaystyle P'(1)=0$ because 1 is a repeated root of $\displaystyle P(x)$ , so it must be a root of P'(x)

and $\displaystyle P(1)=0$ because 1 is a root of P(x)

5. Oh! Silly me. I've forgotten that the apostrophe sign was also a sign for differentiation. Thanks great math!