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# For what value of c will the polynomial P(x)=-2x^3+cx^2-5x+2 have the same remainder when it is divided by x-2 and x+1?

Jan 8, 2017

$c = 11$

#### Explanation:

The remainder theorem states that when a polynomial $p \left(x\right)$ is divided by $x - a$, the remainder is given by $p \left(a\right)$.

Thus, the remainder when $P \left(x\right)$ is divide by $x - 2$ is $- 2 {\left(2\right)}^{3} + c {\left(2\right)}^{2} - 5 \left(2\right) + 2 = - 16 + 4 c - 10 + 2 = - 24 + 4 c$

The remainder when $P \left(x\right)$ is divided by $x + 1$ is $- 2 {\left(- 1\right)}^{3} + c {\left(- 1\right)}^{2} - 5 \left(- 1\right) + 2 = 2 + c + 5 + 2 = 9 + c$

Setting the two as being equal, we have:

$- 24 + 4 c = 9 + c$

$3 c = 33$

$c = 11$

Hopefully this helps!