Question

how do you find the polynomial function that has given zeros and a certain condition. EX: -3, 1, and 4; f(2)=30

Answer 1

\(\bf x=-3\implies x+3=0\implies {\color{blue}{ (x+3)}}=0
\\ \quad \\
x=1\implies x-1=0\implies {\color{blue}{ (x-1)}}=0
\\ \quad \\
x=4\implies x-4=0\implies {\color{blue}{ (x-4)}}=0
\\ \quad \\
{\color{red}{ a}}{\color{blue}{ (x+3)}}{\color{blue}{ (x-1)}}{\color{blue}{ (x-4)}}=0\\ \quad \\\implies {\color{red}{ a}}{\color{blue}{ (x+3)}}{\color{blue}{ (x-1)}}{\color{blue}{ (x-4)}}=\textit{original polynomial}\)

Answer 2

so the function will be say like \(\bf y={\color{red}{ a}}{\color{blue}{ (x+3)}}{\color{blue}{ (x-1)}}{\color{blue}{ (x-4)}}
\\ \quad \\
f(2)=30\implies x=2\qquad y=30\qquad thus
\\ \quad \\
30={\color{red}{ a}}{\color{blue}{ (2+3)}}{\color{blue}{ (2-1)}}{\color{blue}{ (2-4)}}\)

solve for "a"

once you get "a", plug it in the expanded equation