# The polynomial of degree 4, P(x) has a root multiplicity 2 at x=4 and roots multiplicity 1 at x=0 and x=-4 and it goes through the point (5, 18) how do you find a formula for p(x)?

Oct 27, 2016

The polynomial is $P \left(x\right) = \frac{2}{5} x {\left(x - 4\right)}^{2} \left(x + 4\right)$

#### Explanation:

If the polynomial has a root of multiplicity 2 at $x = 4$, the ${\left(x - 4\right)}^{2}$
is a factor

Multiplicity 1 at $x = 0$, then $x$ is a factor

Multiplicity 1 at $x = - 4$, then $\left(x + 4\right)$ is a factor

So $P \left(x\right) = A x {\left(x - 4\right)}^{2} \left(x + 4\right)$

As it pases through $\left(5 , 18\right)$ so
$18 = A \cdot 5 \cdot {\left(5 - 4\right)}^{2} \cdot \left(5 + 4\right)$

So $A = \frac{18}{5} \cdot \frac{1}{9} = \frac{2}{5}$

The polynomial is $P \left(x\right) = \frac{2}{5} x {\left(x - 4\right)}^{2} \left(x + 4\right)$