To fully factor the polynomial \( x^4 - 7x^3 + 9x^2 + 27x - 54 \), we will use a combination of synthetic division, trial and error with possible rational roots, and factoring techniques.

### Step 1: Apply the Rational Root Theorem
The Rational Root Theorem suggests that any possible rational roots are factors of the constant term (−54) divided by factors of the leading coefficient (1).

Factors of \(-54\): \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54 \)
Factors of 1: \( \pm 1 \)

So, the possible rational roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54
\]

### Step 2: Test possible roots
We test these values by substituting them into the polynomial to see if they are roots (i.e., make the polynomial equal to zero).

- **Test \( x = -2 \):**

Substitute \( x = -2 \) into the polynomial:
\[
(-2)^4 - 7(-2)^3 + 9(-2)^2 + 27(-2) - 54 = 16 + 56 + 36 - 54 - 54 = 0
\]
So, \( x = -2 \) is a root.

### Step 3: Divide the polynomial by \( (x + 2) \)
Now that we know \( x = -2 \) is a root, we can divide the polynomial \( x^4 - 7x^3 + 9x^2 + 27x - 54 \) by \( (x + 2) \) using synthetic division or long division.

Using synthetic division:
\[
\begin{array}{r|rrrrr}
-2 & 1 & -7 & 9  & 27 & -54 \\
   &   & -2 & 18 & -54 & 54 \\
\hline
   & 1 & -9 & 27 & -27 & 0 \\
\end{array}
\]
The quotient is \( x^3 - 9x^2 + 27x - 27 \).

### Step 4: Factor the quotient \( x^3 - 9x^2 + 27x - 27 \)
We now factor the cubic \( x^3 - 9x^2 + 27x - 27 \). Notice that this is a factorable expression using the common factor \( (x - 3) \).

Factor out \( (x - 3) \):
\[
x^3 - 9x^2 + 27x - 27 = (x - 3)(x^2 - 6x + 9)
\]

### Step 5: Factor \( x^2 - 6x + 9 \)
The quadratic \( x^2 - 6x + 9 \) is a perfect square trinomial:
\[
x^2 - 6x + 9 = (x - 3)^2
\]

### Step 6: Combine all factors
Thus, the fully factored form of the polynomial is:
\[
x^4 - 7x^3 + 9x^2 + 27x - 54 = (x + 2)(x - 3)^3
\]

### Final Answer:
\[
x^4 - 7x^3 + 9x^2 + 27x - 54 = (x + 2)(x - 3)^3
\]

Would you like further details or explanations? 

Here are some related questions you might find useful:
1. How do you perform synthetic division step-by-step?
2. How does the Rational Root Theorem work?
3. What are perfect square trinomials and how do you recognize them?
4. How can you factor cubic polynomials more generally?
5. Can you factor other polynomials using similar methods?

**Tip:** Always check for common factors and possible rational roots when starting to factor any polynomial.

Discover how to fully factor the polynomial x^4 - 7x^3 + 9x^2 + 27x - 54 using synthetic division, the Rational Root Theorem, and perfect square trinomial factoring techniques. This problem is ideal for high school students in Grades 10-12 learning advanced algebraic methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation