To find the roots of the polynomial \( p(x) = x^3 + 4x^2 - 7x - 10 \), we can use the Rational Root Theorem to test for potential rational roots, or numerical methods or synthetic division to find and factor the polynomial.
Let's start by testing some potential rational roots:
- Possible rational roots can be the factors of the constant term (-10) over the leading coefficient (1). Therefore, the potential rational roots are: \( \pm 1, \pm 2, \pm 5, \pm 10 \).
Testing these values:
- For \( x = -5 \): \[ p(-5) = (-5)^3 + 4(-5)^2 - 7(-5) - 10 = -125 + 100 + 35 - 10 = 0 \] So, \( x = -5 \) is a root.
Now we can divide the polynomial \( p(x) \) by \( x + 5 \) using synthetic division:
\[ \begin{array}{r|rrrr} -5 & 1 & 4 & -7 & -10 \ & & -5 & 5 & 10 \ \hline & 1 & -1 & -2 & 0 \ \end{array} \]
This gives us \( p(x) = (x + 5)(x^2 - x - 2) \).
Next, we need to factor the quadratic \( x^2 - x - 2 \). We can do this by finding the roots of the quadratic using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -1, c = -2 \).
Calculating the discriminant: \[ b^2 - 4ac = (-1)^2 - 4(1)(-2) = 1 + 8 = 9 \]
Now applying the quadratic formula: \[ x = \frac{1 \pm \sqrt{9}}{2} = \frac{1 \pm 3}{2} \]
This gives us: \[ x = \frac{4}{2} = 2 \quad \text{and} \quad x = \frac{-2}{2} = -1. \]
Thus, the roots of \( p(x) = x^3 + 4x^2 - 7x - 10 \) are:
- \( x = -5 \)
- \( x = -1 \)
- \( x = 2 \)
Arranging the roots in ascending order gives us:
The roots are: \[ \boxed{-5, -1, 2. \]