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)

The roots are,_,_ and_

Answer 1

The roots of the polynomial p(x) = x^3 + 4x^2 - 7x - 10 can be found by factoring or by using the Rational Root Theorem.

Using the Rational Root Theorem, we can consider all possible rational roots that are factors of the constant term (10) divided by factors of the leading coefficient (1). The factors of 10 are ±1, ±2, ±5, ±10, and the factors of 1 are ±1.

Checking these possible rational roots in the polynomial, we find that the root x = -2 is a solution.

Now we can perform synthetic division to find the quotient of dividing the given polynomial by (x + 2):

-2 | 1 4 -7 -10
| -2 -4
_______________
1 2 -9 -14

The quotient is x^2 + 2x - 9 with a remainder of -14.

Now we have a quadratic equation, x^2 + 2x - 9, which can be factored as (x + 3)(x - 3).

Therefore, the roots of p(x) = x^3 + 4x^2 - 7x - 10 are x = -2, x = -3, and x = 3 in ascending order.