$$\begin{array}{c|c} x & f(x) \\\hline 0 & 3 \\ 2 & 1 \\ 4 & 0
\\ 5 & -2 \end{array}$$The function $f$ is defined by a polynomial. Some values of $x$ and
$f(x)$ are shown in the above. What must be a factor of $f(x)$?
The question is from a SAT practice test.
I'm not entirely sure what the question is asking.
At first I thought that the $x$ table values could be substituted in the equation and checked if they get the respective $f(x)$ table values. For example, in the equation $x-3$, if $0$ is $x$ then $f(x)$ should be $-3$. However, that doesn't work for any of the equations.
Then I thought, that the $x$ table values must be multiplied by the equations to get $f(x)$, for example, in the equation $x-3$ multiplied by $0$ if $x$ is $0$ to get $0(0-3)$, however that clearly doesn't work either.
So basically, I'm not sure where to start.
According to the answer key, the answer is $(x-4)$, however I don't understand how to get to that answer, hence can anyone please provide a step by step explanation and solution to the problem.
Best Answer
The problem definition assumes that the polynomial $f(x)$ can be factored into $$ f(x) = g(x) h(x) $$ where $g(x)$ and $h(x)$ are polynomials. Furthermore, it can be shown that if $x=a$ is a root for the polynomial, then $x-a$ is a factor for it. This makes sense, since substituting, for ex. $h(x) = x-a$ gives us $$ f(x) = g(x) (x-a) $$ which clearly has a root at $x=a$ ($f(a)= 0$). Reading the table, the polynomial has a root at $x=4$, so therefore $x-4$ is a factor of $f(x)$.