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 given polynomial, we can use the Rational Root Theorem to narrow down the possibilities.

The Rational Root Theorem states that if a rational number (p/q) is a root of a polynomial in the form of p(x) = ax^n + ... + f(x), then p must be a factor of f and q must be a factor of a.

In this case, the leading coefficient is 1, so the possible rational roots are integers that are factors of -10. Possible integer factors of -10 are ±1, ±2, ±5, and ±10.

We can try these values to see if any of them are roots of the polynomial.
For p(x) = x^3+4x^2−7x−10:

When x = -1, p(-1) = (-1)^3+4(-1)^2−7(-1)−10 = -1 + 4 + 7 - 10 = 0.

Therefore, x = -1 is a root of the polynomial.

Using synthetic division, we can divide the polynomial by (x + 1) to find the other roots:

-1 | 1 4 -7 -10
| -1 -3 10
-------------------
1 3 -10 0

The resulting quadratic is x^2 + 3x - 10 = 0.

To solve this quadratic equation, we can use factoring or the quadratic formula.

Factoring x^2 + 3x - 10 = 0, we have (x + 5)(x - 2) = 0.

Setting each factor to 0, we get x + 5 = 0 and x - 2 = 0.
Solving for x, we find x = -5 and x = 2.

Therefore, the roots of p(x) = x^3 + 4x^2 - 7x - 10 are x = -1, x = -5, and x = 2.

The exact answers in ascending order are:
x = -5, -1, 2.