Question

What are the roots of p(x) = x^3 + 4x^2 - 7x- 10. Provide the exact answers. Enter the answers in ascending order from the smallest to the largest number. (2 points)

Answer 1

To find the roots of the polynomial p(x) = x^3 + 4x^2 - 7x - 10, we can use synthetic division, the rational root theorem, or factoring.

The rational root theorem states that if a rational number p/q is a root of the polynomial, where p and q are coprime integers, then p must be a factor of the constant term (-10) and q must be a factor of the leading coefficient (1). Therefore, the possible rational roots to test are ±1, ±2, ±5, and ±10.

Testing the possible roots by synthetic division, we find:

Testing x = -1:
-1 | 1 4 -7 -10
| -1 -3 10
| 1 3 -10 0

Testing x = 2:
2 | 1 4 -7 -10
| 2 12 10
| 1 6 5 0

Testing x = -2:
-2 | 1 4 -7 -10
| -2 -4 22
| 1 2 -11 12

Testing x = 5:
5 | 1 4 -7 -10
| 5 45 190
| 1 9 38 180

From these results, we can see that none of the possible rational roots -1, 2, -2, or 5 is a root of the polynomial p(x) = x^3 + 4x^2 - 7x - 10.

Next, let's try factoring the polynomial. We can try grouping or factoring by inspection. Unfortunately, the polynomial does not appear to have any easily identifiable factors.

Therefore, we can conclude that the polynomial p(x) = x^3 + 4x^2 - 7x - 10 does not have any rational roots or factoring that can be expressed exactly.

To find the approximate roots, we can use numerical methods such as the Newton-Raphson method or bisection method.