# Thread: Polynomial Degrees and Zeros

1. ## Polynomial Degrees and Zeros

Find a polynomial of degree 3 whose zeros are -3, 3/2 and 2

a.2x^3-15x-18
b.2x^2+3x-9
c.2x^2-7x+6
d.2x^3-x^2-15x+18
e.2x^3-7x^2-15x+18

I know C and B are not options because the degree is not three, but how do I figure out the rest?

2. Do you know about the factorial decomposition of polynomials?
If a polynomial $\displaystyle f(x)$ of degree 3 has 3 real roots, letīs say $\displaystyle \alpha, \beta,\gamma$ then you can write it down as $\displaystyle a(x-\alpha)(x-\beta)(x-\gamma)$ where $\displaystyle a$ is the leading coefficient of your polynomial.
Obviously, this fact gets generalised for any polynomial of degree n with m roots, $\displaystyle n<m$.

That means that you can write your polynomial as $\displaystyle a(x+3)(x-3/2)(x-2)$ since they didnīt tell you explicitly nothing about the leading coefficient of the polynomial, $\displaystyle a$ might be any real number different of zero. So set $\displaystyle a=2$ since all your options have L.C.=2.
Apply distributive law (if you have trouble with that tell me and Iīll write it down) and you get option d.