Which of the following intervals *must *contain a root of

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

- –1 <
*x*< 1 - 0 <
*x*< 2 - 1 <
*x*< 3

A. I only

B. II only

C. III only

D. I and II only

E. II and III only

### 1 Answer | Add Yours

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.

**Sources:**

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