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

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 to identify rational roots, followed by solving for the remaining roots.

Testing possible rational roots, we can try \( x = -2 \):

\[
p(-2) = (-2)^3 + 4(-2)^2 - 7(-2) - 10
\]
\[
= -8 + 16 + 14 - 10
\]
\[
= 12 \quad (\text{not a root})
\]

Next, let's try \( x = -1 \):

\[
p(-1) = (-1)^3 + 4(-1)^2 - 7(-1) - 10
\]
\[
= -1 + 4 + 7 - 10
\]
\[
= 0 \quad (\text{is a root})
\]

Since \( x = -1 \) is a root, we can perform synthetic division of \( p(x) \) by \( x + 1 \):

\[
\begin{array}{r|rrrr}
-1 & 1 & 4 & -7 & -10 \\
& & -1 & -3 & 10 \\
\hline
& 1 & 3 & -10 & 0 \\
\end{array}
\]

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

Next, we need to factor \( x^2 + 3x - 10 \). To find the roots of this quadratic equation, we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 3 \), and \( c = -10 \):

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-10)}}{2(1)}
\]
\[
= \frac{-3 \pm \sqrt{9 + 40}}{2}
\]
\[
= \frac{-3 \pm \sqrt{49}}{2}
\]
\[
= \frac{-3 \pm 7}{2}
\]

Calculating the two roots:

1. \( x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \)
2. \( x = \frac{-3 - 7}{2} = \frac{-10}{2} = -5 \)

Thus, the roots of the original polynomial are \( x = -5, -1, 2 \).

Arranging these in ascending order, we have:

\[
\text{The roots are: } -5, -1, \text{ and } 2.
\]

So, the final answer is:

The roots are \(-5\), \(-1\), and \(2\).