Polynomial Function

Algebra Level 3

A third-degree polynomial $P(x)$ , satisfies $P(0)= -3$ and $P(1)= 4$ . When $P(x)$ is divided by $x^{2}+ x+ 1$ , the remainder is $2x -1$ . What is the quotient when $P(x)$ is divided by $x^2 + x + 1$ ?

x-6

3x- 2

x -2

3x-1

1 solution

Chew-Seong Cheong
Apr 28, 2016

Let the quotient be $Q(x) =ax+b$ , then we have:

$\begin{aligned} P(x) & =(x^2 +x+1)(ax+b)+2x-1 \\ P(0) & =(1)(b)-1 \\ - 3&=b-1 \\ \implies b & =-2 \\ P(1 ) & =(3)(a-2)+2-1 \\ 4&=3a-5 \\ \implies a&=3 \end{aligned}$

Therefore, $Q(x) =\boxed {3x-2}$

The solution seems to be formatted incorrectly.

Krutarth Patel - 1 month, 3 weeks ago

Thanks. I think it was caused by the changing of LaTex version.

Chew-Seong Cheong - 1 month, 3 weeks ago

I can see it has been updated. Thanks for the great solution.

Krutarth Patel - 1 month, 3 weeks ago

Polynomial Function

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