Write the polynomial in factored form.  *P*(*x*) = *x*3 + 6*x*2 − 32

To factor the polynomial \( P(x) = x^3 + 6x^2 - 32 \), follow these steps:

### 1. Look for common factors.
There are no common factors among all the terms, so we proceed to other factoring techniques.

### 2. Try grouping or synthetic division.
Since it's a cubic polynomial, we can check for potential rational roots using the Rational Root Theorem. The possible rational roots are factors of the constant term (-32) divided by factors of the leading coefficient (1). These factors are:

\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32
\]

Now, we test some values of \(x\).

- For \( x = 2 \):
\[
P(2) = (2)^3 + 6(2)^2 - 32 = 8 + 24 - 32 = 0
\]
So, \( x = 2 \) is a root.

### 3. Use synthetic division to factor the polynomial.
We divide \( x^3 + 6x^2 - 32 \) by \( x - 2 \) using synthetic division:

\[
\begin{array}{r|rrrr}
2 & 1 & 6 & 0 & -32 \\
  &   & 2 & 16 & 32 \\
\hline
  & 1 & 8 & 16 & 0 \\
\end{array}
\]

The quotient is \( x^2 + 8x + 16 \), and the remainder is 0.

### 4. Factor the quadratic.
Now, we factor the quadratic \( x^2 + 8x + 16 \):

\[
x^2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)^2
\]

### 5. Write the factored form.
Thus, the factored form of \( P(x) = x^3 + 6x^2 - 32 \) is:

\[
P(x) = (x - 2)(x + 4)^2
\]

Would you like further clarification or details on any step?

Here are five related questions to consider:
1. How can the Rational Root Theorem help in factoring other cubic polynomials?
2. What is synthetic division, and when is it useful?
3. How can you check if a polynomial is factored completely?
4. Why does the factored form include \( (x + 4)^2 \)?
5. Can this method be applied to polynomials of degree higher than 3?

**Tip:** Always verify potential roots by substituting them back into the original polynomial to check if they satisfy the equation.

Write the polynomial in factored form. P(x) = x^3 + 6x^2 − 32

Learn how to factor the cubic polynomial P(x) = x^3 + 6x^2 - 32 step-by-step. This guide covers the use of the Rational Root Theorem and synthetic division to simplify polynomials. Suitable for high school students in Grades 9-12.

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