Answer:
[tex]-210[/tex]
Step-by-step explanation:
We have the function:
[tex]f(x)=x^3-2x^2-51x-108[/tex]
And we want to find the remainder when it is divided by [tex](x-2)[/tex].
We can use the Polynomial Remainder Theorem, where if we have a function [tex]f(x)[/tex] divided by a binomial [tex](x-a)[/tex], then our remainder will be [tex]f(a)[/tex].
Here, our divisor is [tex](x-2)[/tex], so our a is 2.
Therefore, our remainder will be [tex]f(2)[/tex]. Let's substitute this for x. This yields:
[tex]f(2)=(2)^3-2(2)^2-51(2)-108[/tex]
Evaluate. Do the exponents:
[tex]f(2)=8-2(4)-51(2)-108[/tex]
Multiply:
[tex]f(2)=8-8-102-108[/tex]
Subtract:
[tex]f(2)=-210[/tex]
So, our remainder is -210.
And we're done!
