Question

What are the roots of p(x)=x3+4x2−7x−10%0D%0A𝑝%0D%0A(%0D%0A𝑥%0D%0A)%0D%0A=%0D%0A𝑥%0D%0A3%0D%0A+%0D%0A4%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A7%0D%0A𝑥%0D%0A−%0D%0A10%0D%0A. 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 attempt to identify possible rational roots using the Rational Root Theorem, synthetic division, or numerical methods.

The Rational Root Theorem suggests testing the factors of the constant term (-10) against the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 2, \pm 5, \pm 10 \).

Let's start testing these values:

1. **Testing \( x = 1 \):**
\[
p(1) = 1^3 + 4(1^2) - 7(1) - 10 = 1 + 4 - 7 - 10 = -12 \quad (\text{not a root})
\]

2. **Testing \( x = -1 \):**
\[
p(-1) = (-1)^3 + 4(-1)^2 - 7(-1) - 10 = -1 + 4 + 7 - 10 = 0 \quad (\text{root found})
\]

Now that we have \( x = -1 \) as a root, we'll use synthetic division to factor the polynomial by \( x + 1 \):

Perform synthetic division:

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

The result is \( x^2 + 3x - 10 \).

Now, we factor \( x^2 + 3x - 10 \):
\[
x^2 + 3x - 10 = (x + 5)(x - 2)
\]

The complete factorization of \( p(x) \) is:
\[
p(x) = (x + 1)(x + 5)(x - 2)
\]

Thus, the roots are:
1. \( x = -1 \)
2. \( x = -5 \)
3. \( x = 2 \)

Ordering these from smallest to largest gives:
\[
-5, -1, 2
\]

Final answers:
\[
\boxed{-5, -1, 2}
\]