A HORRIBLE Remainder Theorem Problem

[Guest]

[tex]P(x)[/tex][tex]\equiv[/tex] [tex]P_{n }x^n + P_{n-1 }x^{n-1}+ ...+P_{0 }[/tex] is divided by [tex](x - a)[/tex], show that the remainder is [tex]P(a)[/tex]
Please show all the steps of your proof.
Thanking you in advance.

[HallsofIvy]

You think this is "HORRIBLE"? I've seen much worse!

Any polynomial, P(x), divided by x- a, has a "quotient" and "remainder": [tex]\frac{P(x)}{x- a}= Q(x)+ \frac{r}{x- a}[/tex]. Since x- a is linear, the quotient, Q, has degree one less than P and r is a constant. From that P(x)= Q(x)(x- a)+ r. Setting x= a, P(a)= Q(a)(0)+ r= r.