# How to write an equation for a rational function with: x intercepts at x = 6 and x = 5?

Dec 30, 2015

Start from the factored form to find

$f \left(x\right) = {x}^{2} - 11 x + 30$

#### Explanation:

An $x$ intercept occurs when $f \left(x\right) = 0$. Then, we are just looking for a rational function $f \left(x\right)$ such that $f \left(6\right) = f \left(5\right) = 0$. We can easily construct a polynomial with the desired property by starting in factored form.

It should be clear that if $f \left(x\right) = \left(x - 6\right) \left(x - 5\right)$ that plugging in $5$ or $6$ will give $0$. Then, as a polynomial is certainly a rational function, it satisfies the desired requirements, and so we can just expand to get

$f \left(x\right) = {x}^{2} - 11 x + 30$