Find integer bounds for the roots of the following equation. $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$
A) $- 3,6$
B) $- 2,4$
C) $- 3,3$
D) 0,3
E) none of these

Question

Quizplus · Accepted Answer

We can use the Rational Root Theorem to narrow down the possible integer roots of the polynomial. The potential rational roots are $\pm 1, \pm 3, \pm 5, \pm 15$. Testing these values, we find that $x=3$ is a root, and using polynomial division or synthetic division, we see that $(3x-2)$ is a factor of the polynomial. Thus, the other roots can be found by solving:
$$3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = (3x-2)(x^4-3x^3-3x^2+11x+8)$$
The second factor can be factored as $(x-1)(x+4)(x^2-x-2)$ using either further testing or the rational root theorem again. The possible integral roots of the second factor are $\pm 1, \pm 2, \pm 4, \pm 8$. Testing these values, we find that the only integer roots are $x=1$ and $x=-2$. Thus, the integer roots of the original equation are $x=-2,1,3$. The only choice with all these roots is (B).

Find Integer Bounds for the Roots of the Following Equation 3x5−11x4−2x3+38x2−21x−15=03 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 03x5−11x4−2x3+38x2−21x−15=0

Find Integer Bounds for the Roots of the Following Equation $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$