To factorize \( f(x) = x^4 + 4x^3 - 7x^2 - 22x + 24 \), we can proceed by trying to find its roots and then using polynomial division or synthetic division.

### Step 1: Try Potential Rational Roots
Using the Rational Root Theorem, the possible rational roots of the polynomial are the divisors of the constant term \(24\) divided by the divisors of the leading coefficient \(1\). Therefore, the possible rational roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24
\]

### Step 2: Test for Rational Roots
Testing some of these values, we find:
- \( f(2) = 2^4 + 4 \cdot 2^3 - 7 \cdot 2^2 - 22 \cdot 2 + 24 = 0 \)
- So, \( x = 2 \) is a root.

### Step 3: Use Synthetic Division
Since \( x = 2 \) is a root, we can divide \( f(x) \) by \( x - 2 \) using synthetic division:

\[
\begin{array}{r|rrrrr}
2 & 1 & 4 & -7 & -22 & 24 \\
  &   & 2 & 12 & 10 & -24 \\
\hline
  & 1 & 6 & 5 & -12 & 0 \\
\end{array}
\]

This gives the quotient \( x^3 + 6x^2 + 5x - 12 \).

### Step 4: Factor the Quotient Polynomial
Now, we have:
\[
f(x) = (x - 2)(x^3 + 6x^2 + 5x - 12)
\]
Next, we need to factor \( x^3 + 6x^2 + 5x - 12 \).

Testing more roots, we find that \( x = -3 \) is also a root of \( x^3 + 6x^2 + 5x - 12 \).

### Step 5: Synthetic Division on Quotient Polynomial
We divide \( x^3 + 6x^2 + 5x - 12 \) by \( x + 3 \) using synthetic division:

\[
\begin{array}{r|rrrr}
-3 & 1 & 6 & 5 & -12 \\
   &   & -3 & -9 & 12 \\
\hline
   & 1 & 3 & -4 & 0 \\
\end{array}
\]

This gives the quotient \( x^2 + 3x - 4 \).

### Step 6: Factor the Quadratic
Now we have:
\[
f(x) = (x - 2)(x + 3)(x^2 + 3x - 4)
\]
The quadratic \( x^2 + 3x - 4 \) can be factored further as:
\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\]

### Final Answer
Thus, the complete factorization of \( f(x) \) is:
\[
f(x) = (x - 2)(x + 3)(x + 4)(x - 1)
\]

Would you like details on any steps, or have questions?

---

1. How is the Rational Root Theorem used in factoring polynomials?
2. Can every polynomial be factored using synthetic division?
3. What are other methods for factoring higher-degree polynomials?
4. How do you verify if a factorization is correct?
5. What are some practical applications of polynomial factorization?

**Tip:** Testing rational roots early can simplify the factorization process significantly.

Learn to factorize the polynomial x^4 + 4x^3 - 7x^2 - 22x + 24 with detailed steps. This guide explores key algebraic techniques, including the Rational Root Theorem and synthetic division, making it ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation