# How do you find the polynomial function with leading coefficient 2 that has the given degree and zeroes: degree 3: zeroes 2, 1/2, 3/2?

Sep 4, 2016

The polynomial function is
$2 {x}^{3} - 8 {x}^{2} + \frac{19}{2} x - 3$

#### Explanation:

A polynomial function with zeros as $a$, $b$ and $c$ can be written as

$\left(x - a\right) \left(x - b\right) \left(x - c\right)$

However, this will give the leading coefficient as $1$. If leading coefficient is $n$, the polynomial would be $n \left(x - a\right) \left(x - b\right) \left(x - c\right)$.

Hence a polynomial function of degree $3$, zeros $2$, $\frac{1}{2}$ and $\frac{3}{2}$ and leading coefficient $2$ is

$2 \left(x - 2\right) \left(x - \frac{1}{2}\right) \left(x - \frac{3}{2}\right)$

= $2 \left(x - 2\right) \left({x}^{2} - 2 x + \frac{3}{4}\right)$

= $2 \left({x}^{3} - 2 {x}^{2} + \frac{3}{4} x - 2 {x}^{2} + 4 x - \frac{3}{2}\right)$

= $2 {x}^{3} - 8 {x}^{2} + \frac{19}{2} x - 3$