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
Accepted Solution
A:
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
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
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: