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Which of the following intervals must contain a root of 2x^3-x^2-x -3 = 0 –1 <x...

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christiano-cr7 | Salutatorian

Posted September 25, 2013 at 5:31 AM via web

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Which of the following intervals must contain a root of

2x^3-x^2-x -3 = 0

  1. –1 <x < 1
  2. 0 <x < 2
  3.  1 <x < 3

A. I only

B. II only

C. III only

D. I and II only

E. II and III only

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llltkl | College Teacher | Valedictorian

Posted September 25, 2013 at 6:13 AM (Answer #1)

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This is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Theorem.

The Rational Root Theorem tells that if the polynomial has a rational zero then it must be a fraction` p/q` , where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factors of the leading coefficient (2) are 1 and 2 .The factors of the constant term (-3) are 1 and 3 . Then the Rational Roots Tests yields the following possible solutions:

`+-1/1, +-1/2, +-3/1, +-3/2`

S12ubstitute the possibe roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the given polynomial `P(x)` , `2x^3-x^2-x -3 = 0` , we obtain `P(3/2)=0`

Hence, one of the roots of the given polynomial is `3/2` which lies in the intervals `0 ltx lt 2` and `1 ltx lt 3` .

Therefore, the correct answer is option E.

To find remaining zeros we use Factor Theorem.

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