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Find all roots of x^3-2x^2-5x+6

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Find all roots of x^3-2x^2-5x+6.

asked Feb 24, 2014 in ALGEBRA 2 by homeworkhelp Mentor

2 Answers

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Given polynomial x^3-2x^2-5x+6

By synthetic division

image

x^3-2x^2-5x+6 = (x-1)(x+2)(x-3)

Roots are x = 1,-2,3.

answered Feb 24, 2014 by david Expert
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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- 5x + 6 = 0.

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 1.

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

The possible values for q are ± 1.

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

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

- 5

6

- 1

1

- 3

8

- 2

2

1

0 - 5 - 4

- 3

1

- 5 10 - 9

1

1 - 1 - 6 0

Since, f(1) = 0, x = 1 is a zero. The depressed polynomial is  x 2 - x - 6 = 0.

By factor by grouping.

x 2 - 3x + 2x - 6 = 0

x(x - 3) + 2(x - 3) = 0

Factor : (x - 3)(x + 2) = 0

Apply zero product property.

x - 3 = 0 and x + 2 = 0

x = 3 and x = - 2.

Therefore, tthe roots of the function are x = 1, x = - 2, and x = 3.

answered May 15, 2014 by lilly Expert

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