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

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

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Answer 1 · 2016-09-10T17:13:51

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 $\pm 1$ $+-1$

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

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

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.

$\frac{1}{3} ⌉ 30 a a - 1 a a a - 6 a a a a 1$ $1/3~|30color(white)(aa)-1color(white)(aaa)-6color(white)(aaaa)1$
$a a a a a a a a a 10 a a a a a 3 1 a - 1$ $color(white)(aaaaaaaaa)10color(white)(aaaaa)3color(white)(1a)-1$
$a a a$ $color(white)(aaa)$ _________

$a a a 30 a a a a a 9 a a a - 3 a a a a 0$ $color(white)(aaa)30color(white)(aaaaa)9color(white)(aaa)-3color(white)(aaaa)0$

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

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

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

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

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

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Explanation:

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What are all the possible rational zeros for y=30x3−x2−6x+1y=30x^3-x^2-6x+1 and how do you find all zeros?

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