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when the function g(x)=x³+bx²-5x+2 is divided by (x-1) it leaves the same remainder as when it is divided by (x+2). Find the value of b.

I ve tried this so many times and I get -2 as my answer but it says 2 in the back. Some one please help

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Hint: The question is equivalent to

Find $b$ given that $g(1)=g(-2)$.

This follows from this general fact:

The remainder of a polynomial $f(x)$ divided by $x-a$ is $f(a)$.

So, we have to solve $$ 1+b-5+2=-8+4b+10+2 $$ for $b$. This gives $b=-2$, as you have found.

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  • $\begingroup$ It appears the OP knows this since OP got the correct answer $b=-2$ whereas the book answer was incorrect. $\endgroup$ – John Wayland Bales Feb 7 '17 at 15:08
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Your answer is correct, the book is wrong.

\begin{eqnarray} x^3-2x^2-5x+2&=&(x-1)(x^2-x-6)-4\\ &=&(x+2)(x^2-4x+3)-4 \end{eqnarray}

with equal remainders of $-4$.

Whereas

\begin{eqnarray} x^3+2x^2-5x+2&=&(x-1)(x^2+3x-0)+0\\ &=&(x+2)(x^2+4x+3)+8 \end{eqnarray}

which have unequal remainders of $0$ and $8$.

So the correct answer is $b=-2$.

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  • $\begingroup$ Does this answer your question? $\endgroup$ – John Wayland Bales Feb 7 '17 at 15:02

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