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)? −881224

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

[tex]\dfrac{p(x)}{x-c}=q(x)+\dfrac{r(x)}{x-c}\iff p(x)=(x-c)q(x)+r(x)[/tex]

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

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

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

Since [tex]q(x)[/tex] is a polynomial (not a rational expression), then we know that [tex]x+1[/tex] divides [tex]x^3-4x^2+15x+k+8[/tex] exactly. In particular, the remainder term of this quotient is 0. We can use long or synthetic division to determine [tex]q(x)[/tex]. 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 [tex]k-12=0\implies k=12[/tex].

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