https://file.aiquickdraw.com/mathbot/file/172928739183996k2snya.jpg

Let's factor the given polynomial: 

\[
x^3 + 8x^2 + 19x + 12
\]

### Step 1: Find a possible root
Using the Rational Root Theorem, the possible rational roots of the polynomial are the factors of the constant term (12) divided by the factors of the leading coefficient (1). The possible roots are:

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

### Step 2: Test the roots using synthetic division or substitution
Let's test \( x = -1 \):

\[
(-1)^3 + 8(-1)^2 + 19(-1) + 12 = -1 + 8 - 19 + 12 = 0
\]

Since \( x = -1 \) is a root, \( x + 1 \) is a factor.

### Step 3: Perform synthetic division
Now, divide \( x^3 + 8x^2 + 19x + 12 \) by \( x + 1 \) using synthetic division.

\[
\begin{array}{r|rrrr}
-1 & 1 & 8 & 19 & 12 \\
   &   & -1 & -7 & -12 \\
\hline
   & 1 & 7 & 12 & 0 \\
\end{array}
\]

The quotient is \( x^2 + 7x + 12 \).

### Step 4: Factor the quotient
Now, factor \( x^2 + 7x + 12 \):

\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

### Step 5: Combine the factors
Thus, the complete factorization of \( x^3 + 8x^2 + 19x + 12 \) is:

\[
(x + 1)(x + 3)(x + 4)
\]

The correct answer is:

\[
\boxed{A. \ (x + 4)(x + 3)(x + 1)}
\]

Would you like more details on any step or have any questions?

Here are 5 related questions:
1. How do you perform synthetic division on polynomials?
2. What is the Rational Root Theorem and how is it used in factoring?
3. Can all cubic polynomials be factored using rational roots?
4. How do you factor quadratic expressions like \(x^2 + 7x + 12\)?
5. What are some strategies for testing potential roots of polynomials?

**Tip**: Always check for simple roots (like \( x = 1 \) or \( x = -1 \)) first, as they often lead to quicker solutions when factoring polynomials.

What is the complete factorization of the polynomial x^3 + 8x^2 + 19x + 12?

Learn how to completely factor the cubic polynomial x^3 + 8x^2 + 19x + 12. The solution involves testing rational roots, using synthetic division, and factoring quadratic expressions. Ideal for high school students studying polynomial factorization.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation