Question

What are the roots of p(x)=x3+4x2−7x−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

To find the roots of the polynomial \( p(x) = x^3 + 4x^2 - 7x - 10 \), we can apply the Rational Root Theorem and synthetic division, or we can use numerical algorithms or graphical methods.

1. **Finding potential rational roots using the Rational Root Theorem**:
The possible rational roots are factors of the constant term (-10) divided by the factors of the leading coefficient (1). The possible rational roots are:
\[
\pm 1, \pm 2, \pm 5, \pm 10
\]

2. **Testing these potential roots**:

- Let's test \( x = -5 \):
\[
p(-5) = (-5)^3 + 4(-5)^2 - 7(-5) - 10 = -125 + 100 + 35 - 10 = 0
\]
So, \( x = -5 \) is a root.

3. **Performing synthetic division** with \( x + 5 \):
We divide the polynomial \( p(x) \) by \( x + 5 \):
\[
\begin{array}{r|rrrr}
-5 & 1 & 4 & -7 & -10 \\
& & -5 & 5 & 10 \\
\hline
& 1 & -1 & -2 & 0 \\
\end{array}
\]
The quotient is \( x^2 - x - 2 \).

4. **Factoring the quadratic** \( x^2 - x - 2 \):
\[
x^2 - x - 2 = (x - 2)(x + 1)
\]
So we have:
\[
p(x) = (x + 5)(x - 2)(x + 1)
\]

5. **Finding all roots**:
The roots are:
\[
x + 5 = 0 \Rightarrow x = -5,
\]
\[
x - 2 = 0 \Rightarrow x = 2,
\]
\[
x + 1 = 0 \Rightarrow x = -1.
\]

6. **Writing the roots in ascending order**:
The roots are:
\[
-5, -1, \text{ and } 2.
\]

Thus, the final answer is:
The roots are \( -5, -1, \) and \( 2 \).