Problem 1: Find k such that f(x)=x^4-kx^3+kx^2+1 has the factor (x+2)
Problem 2: What is the remainder when f(x)=-3x^17+x^9-X^5+2x is divided by (x+1)
Thank you
Hello, oceanmd!
I believe you're expected to know the Factor Theorem and Remainder Theorem.
Problem 1: Find $\displaystyle k$ such that $\displaystyle f(x)\:=\:x^4-kx^3+kx^2+1$ has the factor $\displaystyle (x+2)$
If $\displaystyle (x+2)$ is a factor of $\displaystyle f(x)$, then: .$\displaystyle f(\text{-}2) \:=\:0$
Hence: .$\displaystyle f(\text{-}2) \;=\;(\text{-}2)^4 - k(\text{-}2)^3 + k(\text{-}2)^2 + 1 \;=\;0$
. . and we have: .$\displaystyle 16 + 8k + 4k + 1 \:=\:0\quad\Rightarrow\quad12k \:=\:-17\quad\Rightarrow\quad\boxed{k \:=\:-\frac{17}{12}}$
When $\displaystyle f(x)$ is divided by $\displaystyle (x+1)$, the remainder is: .$\displaystyle f(\text{-}1)$Problem 2: What is the remainder when $\displaystyle f(x)\:=\:-3x^{17}+x^9-x^5+2x$ is divided by $\displaystyle (x+1)$
. . $\displaystyle f(\text{-}1) \;=\;-3(\text{-}1)^{17} + (\text{-}1)^9 - (\text{-}1)^5 + 2(\text{-}1) \;=\;3 - 1 + 1 - 2 \;=\;\boxed{1}$