Question

WILL MARK BRAINLIEST!!! Given p(x)=x3−4x2+15x+k and the remainder of the division of p(x) by (x+1) is equal to −8, what is the remainder of the division of p(x) by (x−1)?
−8

8

12

24

Answer 1

Use the polynomial remainder theorem. If $p(x)$ is a polynomial of degree $n>1$, then we can divide by a linear term $x-c$ to get a quotient $q(x)$ and remainder $r(x)$ of the form

$\dfrac{p(x)}{x-c}=q(x)+\dfrac{r(x)}{x-c}\iff p(x)=(x-c)q(x)+r(x)$

Then when $x=c$, we get $p(c)=r(c)$. In other words, the value of $p(x)$ at $x=c$ tells you the value of the remainder upon dividing $p(x)$ by $x-c$.

So given that $p(x)=x^3-4x^2+15x+k$, and the remainder upon dividing $p(x)$ by $x+1$ is -8, we know that $r(-1)=-8$, so

$\dfrac{x^3-4x^2+15x+k}{x+1}=q(x)-\dfrac8{x+1}$
$\dfrac{x^3-4x^2+15x+k+8}{x+1}=q(x)$

Since $q(x)$ is a polynomial (not a rational expression), then we know that $x+1$ divides $x^3-4x^2+15x+k+8$ exactly. In particular, the remainder term of this quotient is 0. We can use long or synthetic division to determine $q(x)$. I prefer typing out the work for synthetic division:

-1   |   1   -4   15    k + 8
.     |        -1     5       -20
- - - - - - - - - - - - - - - - - -
.     |   1   -5    20    k - 12

The remainder here has to be 0, so $k-12=0\implies k=12$.

Finally, we can get the remainder upon dividing $p(x)$ by $x-1$ by evaluating $p(1)$, which gives $p(1)=24$.

Answer 2

The answer is 24
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