# Finding Zeros of Cubic Polynomial

• Oct 25th 2012, 10:07 PM
petenice
Finding Zeros of Cubic Polynomial
I am having problems factoring this cubic polynomial. I can't find how they got to the factored form on the right.

However i was able to get to the same answer but first factored, then found possible zeros using p/q, then did long division, then used the quadratic formula. Is there a faster way to do this?

Thanks.
• Oct 25th 2012, 10:37 PM
chiro
Re: Finding Zeros of Cubic Polynomial
Hey petenice.

Factoring is a weird thing: in root finding a wide range of approaches are taken.

One that is commonly taught is to "guess" one root and then factor the polynomial by dividing P(X) by (x-a) where a is the root. It's not a mechanical systematic thing unless you have previous experience or can "see" something in the polynomial itself (i.e. an easy solution).

There is however a formula for obtaining the roots to a cubic polynomial given a cubic equation and it is more complex than the standard quadratic.

You also have other results in algebra that give ways to factorize equations of a particular form.

The cubic formula gives you a systematic way to get the roots and the proof shows you how the routine is constructed, but when you don't have these and try to "guess" roots then that's what happens: you make a few guesses and if you get one right then you do the long division with a polynomial and you factorize out to solve something simpler.
• Oct 25th 2012, 10:48 PM
MarkFL
Re: Finding Zeros of Cubic Polynomial
I would use the rational roots theorem to see if a rational root exists as you did, but I would first factor out a 2 to make things a little simpler.
• Oct 26th 2012, 03:48 AM
Petrus
Re: Finding Zeros of Cubic Polynomial
hello! if u intressted i can show u how to solve all the roots with rational root theorem and polynomal division!:)
• Oct 26th 2012, 04:57 AM
Salahuddin559
Re: Finding Zeros of Cubic Polynomial
Hi, please take a look at: Cubic function - Wikipedia, the free encyclopedia, you can use some of the simpler methods there.

Salahuddin
Maths online
• Oct 26th 2012, 08:16 AM
Soroban
Re: Finding Zeros of Cubic Polynomial
Hello, petenice!

I don't think there is a faster way . . .

We want to solve: . $32x^3 - 48x^2 + 12x + 2 \:=\:0$

Divide by 2: . $16x^3 - 24x^2 + 6x + 1 \:=\:0$

According to the Rational Roots Theorem,
. . the only possible rational roots are: . $\pm1,\:\pm\tfrac{1}{2},\:\pm\tfrac{1}{4},\:\pm \tfrac{1}{8},\:\pm\tfrac{1}{16}$

We find that $x = \tfrac{1}{2}$ is a zero of the polynomial.
. . Hence, $2x-1$ is a factor.

Using long division: . $16x^3 - 24x^2 + 6x + 1 \;=\;(2x-1)(8x^2-8x-1)$

The other two roots arise from: . $8x^2 - 8x - 1 \:=\:0$

. . $x \;=\;\frac{\text{-}(\text{-}8) \pm \sqrt{(\text{-}8)^2 - 4(8)(\text{-}1)}}{2(8)} \;=\;\frac{8 \pm\sqrt{96}}{16} \;=\; \frac{8 \pm4\sqrt{6}}{16} \;=\;\frac{2\pm\sqrt{6}}{4}$