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how do you solve p(x)=3x^4-8x^3-6x^2+17x+6

0 votes
a. list all the possible rational zeros

b. find the rational zeros.
asked Mar 8, 2014 in ALGEBRA 2 by harvy0496 Apprentice

2 Answers

0 votes

The polinomial is P (x ) = 3x4 - 8x3 - 6x2 + 17x + 6.

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation anxn + an  1xn – 1 + ... + a1x + a0 = 0, then p is a factor of a0 and q is a factor if an.

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 3.

The possible values of p are   ± 1, ± 2, ± 3.

The possible values for q are ± 1, ± 3.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 3, ± 1/3, ± 2/3.

Make a table for the synthetic division and test possible real zeros.

p/q

3

- 8

- 6

17

6

1

3

- 2

- 10

- 3

0

2

3

- 2

- 10

- 3

0

Since f(2) = 0, x = 2 is a zero. The depressed polynomial is  3x3 - 2x2 – 10x - 3 = 0.

answered Apr 1, 2014 by lilly Expert
0 votes

Contd.....

If p/q is a rational zero, then p is a factor of 3 and q is a factor of 3.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1,  ± 3, ± 1/3.

Make a table for the synthetic division and test possible real zeros.

p/q

3

- 2

- 10

- 3

1

3

1

- 9

- 12

- 1

3

- 6

- 4

1

-1/3

3

- 3

- 9

0

Since f(- 1/3) = 0, x = - 1/3 is a zero. The depressed polynomial is  3x– 3x - 9 = 0.

Devide each side by 3.

xx - 3 = 0.

Since the depressed polynomial of this zero, xx - 3, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equationxx - 3.

x = [-b ± √(b^2 - 4ac)]/2a

Substitute b = - 1, a = 1, and c = - 3.

x = [-(- 1) ± √((- 1)^2 - 4 * 1 * (- 3))]/2 * 1

x = [ 1 ± √(1 + 12)]/2

x = [ 1 ± √(13)]/2

x = [ 1 ± 3.6]/2

x = [ 1 + 3.6]/2 and x = [ 1 - 3.6]/2

x = 4.6/2 and x = - 2.6/2

x = 2.3 and x = - 1.3 .

The rational zeros of the given polinomail are x = 2, 2.3, - 1.3, and - 1/3.

answered Apr 1, 2014 by lilly Expert

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