Find the value of k

Question

Find the value of k if

$(3x + k)^3 + (4x - 7)^2$ has a remainder of $33$ when divided by $x-3$

Should I split up the main equation: $(3x + k)^3 + (4x - 7)^2$ ?

Answer 1

Hint: Let $P(x)$ be your polynomial. Then $P(3)=33$.

Remark: We are using the Remainder Theorem, which is entirely different from the Chinese Remainder Theorem.

The Remainder Theorem says that if $P(x)$ is a polynomial, then the remainder when you divide $P(x)$ by $x-a$ is $P(a)$.

The proof is straightforward. Let $P(x)=(x-a)Q(x0+r$. Put $x=a$. The $(x-a0Q(x)$ part dies, and we get $r=P(a)$.