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find the roots of the polynomial equation x^3-2x^2+10x+136=0

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I need help figuring out how to find the roots of this polynomial equation.

asked Feb 24, 2014 in ALGEBRA 2 by skylar Apprentice

3 Answers

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Given polynomial equation x^3-2x^2+10x+136 = 0

By synthatic division

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answered Feb 25, 2014 by david Expert
0 votes

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 anx n + an  1x n – 1 + ... + a1x + a0 = 0, then p is a factor of a0 and q is a factor if an.

The function is x 3- 2x 2+ 10x + 136 = 0.

If p/q is a rational zero, then p is a factor of 136 and q is a factor of 1.

The possible values of p are   ± 1,   ± 4, and ± 34.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1,   ± 4, and ± 34.

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

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

p/q

1

- 2

10

136

1

1

- 1

9

145

2

1

0 10 156

3

1

1 13 175

-1

1 - 3 13 123

- 2

1 - 4 18 100

- 3

1 - 5 25 61

- 4

1 - 6 34 0

Since, f(- 4) = 0, x = - 4 is a zero. The depressed polynomial is  x 2 - 6x + 34 = 0

answered May 15, 2014 by lilly Expert
0 votes

Since the depressed polynomial of this zero, x2-2x+5, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

x = [ - b ± √(b 2 - 4ac ) ] / 2a.

Substitute a = 1, b = - 6, and c = 34.

x = [ - (- 6) ± √((- 6)2 - 4 * 1 * 34) ] / 2 * 1

x = [ 6 ± √(36 - 136) ] / 2

x = [ 6 ± √(- 100) ] / 2

x = [ 6 ± 10i ] / 2

⇒ x = 3 ± 5i.

Therefore, the roots of the function are x = - 4, x = 3 + 5i., and x = 3 - 5i.

answered May 15, 2014 by lilly Expert

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