Question

What is a quartic function with only two real zeros, x = 5 and x = 9?

Answer 1

hmmm a quartic, meaning 4 degree
so, 4 answers or solutions, or zeros
so... say
x = 5
x = 5
x = 9
x = 9

so there, meaning \(\begin{array}{llll}
x=5\implies &x-5=0\\
x=5\implies &x-5=0\\
x=9\implies &x-9=0\\
x=9\implies &x-9=0\\
\end{array}
\\ \quad \\
(x-5)(x-5)(x-9)(x-9)=\textit{original polynomial}\)

Answer 2

A quadratic means it has a degree of 2 as in \[ax^2+bx+c\] and since x= 5 and x=9 are it's solutions, then the factored polynomial is \[(x-5)(x-9)\] and the expanded form is \[x^2-14x+45\]