Which of the following intervals must contain a root of 2x^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
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.