# What are all the possible rational zeros for y=30x^3-x^2-6x+1 and how do you find all zeros?

Sep 10, 2016

Use the Rational Zero Theorem, synthetic division, and factoring.

#### Explanation:

According to the Rational Zero Theorem, the list of all possible rational zeros is obtained by dividing all the factors of the constant term 1 by all the factors of the leading coefficient term 30.

The possible factors of 1 are $\pm 1$

The possible factors of 30 are $\pm 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30$

The possible zeros are:
$\pm \frac{1}{1} , \pm \frac{1}{2} , \pm \frac{1}{3} , \pm \frac{1}{5} , \pm \frac{1}{6} , \pm \frac{1}{10} , \pm \frac{1}{15} , \pm \frac{1}{30}$

To find all the zeros, use synthetic division. Pick one of the possible zeros as a divisor. If the remainder is zero, the divisor is a zero. If the remainder is not zero, choose another possible zero and try again. I chose 1/3 because I "cheated" and first checked the zeros using a graphing utility.

$\frac{1}{3} \rceiling 30 \textcolor{w h i t e}{a a} - 1 \textcolor{w h i t e}{a a a} - 6 \textcolor{w h i t e}{a a a a} 1$
$\textcolor{w h i t e}{a a a a a a a a a} 10 \textcolor{w h i t e}{a a a a a} 3 \textcolor{w h i t e}{1 a} - 1$
$\textcolor{w h i t e}{a a a}$_________

$\textcolor{w h i t e}{a a a} 30 \textcolor{w h i t e}{a a a a a} 9 \textcolor{w h i t e}{a a a} - 3 \textcolor{w h i t e}{a a a a} 0$

$\frac{1}{3}$ is a zero because the remainder is zero.

Write the quotient using the coefficients found in synthetic division and set it equal to zero.
$30 {x}^{2} + 9 x - 3 = 0$

Factor and solve to find the remaining zeros:
$3 \left(10 {x}^{2} + 3 x - 1\right) = 0$
$3 \left(5 x - 1\right) \left(2 x + 1\right) = 0$
$x = \frac{1}{5}$ and $x = - \frac{1}{2}$

The three zeros are $x = - \frac{1}{2} , x = \frac{1}{5} , x = \frac{1}{3}$