Write the polynomial of degree $4$ with $x$ intercepts of $(\frac{1}{2},0), (6,0)$ and $(-2,0)$ and $y$ intercept of $(0,18)$. Write the polynomial of degree $4$ with $x$ intercepts of $(\frac{1}{2},0), (6,0) $ and $ (-2,0)$ and $y$ intercept of $(0,18)$.
The root ($\frac{1}{2},0)$ has multiplicity $2$.
I am to write the factored form of the polynomial with the above information. I get:
$f(x)=-6\big(x-\frac{1}{2}\big)^2(x+2)(x-6)$
Whereas the provided solution is:
$f(x)=-\frac{3}{2}(2x-1)^2(x+2)(x-6)$
Here's my working:
Write out in factored form:
$f(x) = a\big(x-\frac{1}{2}\big)^2(x+2)(x-6)$
I know that $f(0)=18$ so:
$$18 = a\big(-\frac{1}{2}\big)^2(2)(-6)$$
$$18 = a\big(\frac{1}{4}\big)(2)(-6)$$
$$18 = -3a$$
$$a = -6$$
Thus my answer:
$f(x)=-6\big(x-\frac{1}{2}\big)^2(x+2)(x-6)$
Where did I go wrong and how can I arrive at:
$f(x)=-\frac{3}{2}(2x-1)^2(x+2)(x-6)$ ?

$$\Big(x-\dfrac{1}{2}\Big)^2 = \Big(\color{red}{\frac{1}{2}}\Big(2x-1\Big)\Big)^2=\color{red}{\frac{1}{4}}\Big(2x-1\Big)^2$$
