https://file.aiquickdraw.com/mathbot/file/1727041187503mnttzp3y.jpg

To write the polynomial \( x^3 + x^2 - 8x - 12 \) as a product of linear factors, we need to factor it fully. Let's proceed step by step:

### Step 1: Look for possible rational roots.
Using the **Rational Root Theorem**, the possible rational roots are factors of the constant term (-12) divided by the factors of the leading coefficient (1), so the possible roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
\]

### Step 2: Test the possible roots using synthetic division or substitution.
Let’s try \( x = 2 \):

\[
f(2) = 2^3 + 2^2 - 8(2) - 12 = 8 + 4 - 16 - 12 = -16
\]

\( x = 2 \) is not a root.

Now, try \( x = -2 \):

\[
f(-2) = (-2)^3 + (-2)^2 - 8(-2) - 12 = -8 + 4 + 16 - 12 = 0
\]

So, \( x = -2 \) is a root.

### Step 3: Perform synthetic division.
We will now divide \( x^3 + x^2 - 8x - 12 \) by \( x + 2 \) using synthetic division:

\[
\begin{array}{r|rrrr}
-2 & 1 & 1 & -8 & -12 \\
   &   & -2 & 2  & 12  \\
\hline
   & 1 & -1 & -6 & 0   \\
\end{array}
\]

The quotient is \( x^2 - x - 6 \).

### Step 4: Factor the quadratic \( x^2 - x - 6 \).
We need to factor \( x^2 - x - 6 \). We find two numbers that multiply to -6 and add to -1. These numbers are -3 and 2, so:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

### Step 5: Write the polynomial as a product of linear factors.
We now have the full factorization:

\[
x^3 + x^2 - 8x - 12 = (x + 2)(x - 3)(x + 2)
\]

Alternatively, this can be written as:

\[
(x + 2)^2(x - 3)
\]

This is the polynomial expressed as a product of linear factors.

Would you like further details or clarifications?

---

Here are some related questions you may want to explore:
1. How does the Rational Root Theorem help in factoring polynomials?
2. Can all cubic polynomials be factored into linear factors?
3. What happens when there are complex roots in the factorization process?
4. How does synthetic division differ from polynomial long division?
5. How can you check if your factorization is correct?

**Tip**: Always check your factorization by multiplying the factors back to ensure you get the original polynomial!

Write the polynomial x^3 + x^2 - 8x - 12 as a product of linear factors.

Learn how to factor the cubic polynomial x^3 + x^2 - 8x - 12 step by step using the Rational Root Theorem and synthetic division. This problem is ideal for students in Grades 10-12 and covers polynomial factoring concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation