How to Factorize (x^4 - x - 3^2): A Comprehensive Guide

In this guide, we will walk you through the process of factorizing the polynomial (x^4 - x - 3^2). This polynomial can be factorized using a combination of algebraic identities and substitution methods. Let's delve into the steps involved and understand the reasoning behind each step.

Introduction to Factorization

Factorization is a fundamental operation in algebra that involves breaking down a polynomial into its constituent factors, which are simpler polynomials whose product equals the original polynomial. This process can be critical in solving equations, simplifying expressions, and understanding the behavior of functions.

Initial Expression

The given expression is (x^4 - x - 3^2). Let's start by writing it down:

(x^4 - x - 9)

Step 1: Recognize the Difference of Squares

The expression (x^4 - 9) can be recognized as a difference of squares. The difference of squares formula is given by:

(a^2 - b^2 (a - b)(a b))

In this case, we have:

(x^4 - 9 x^4 - 3^2)

Here, (a x^2) and (b 3). Applying the difference of squares formula:

(x^4 - 9 (x^2 - 3)(x^2 3))

Step 2: Factorize Further if Possible

Now, we need to factorize the expression (x^4 - x - 9). We already have the factorization of (x^4 - 9) as:

(x^4 - x - 9 (x^2 - 3)(x^2 3) - x)

Step 3: Use Substitution for Simplification

To further simplify the expression, we can use the substitution method. Let's set:

(a x^2) and (b x - 3)

Then, the expression becomes:

(a^2 - b^2 - a)

Using the difference of squares identity:

(a^2 - b^2 (a - b)(a b))

Substitute back:

((x^2 - (x - 3))(x^2 (x - 3)) - x)

Simplify the terms inside the parentheses:

((x^2 - x 3)(x^2 x - 3) - x)

Final Factorization

After simplification, the final factorization of the expression is:

(x^4 - x - 9 (x^2 - x 3)(x^2 x - 3) - x)

Verification

To verify the factorization, we can expand the expression and check if it matches the original polynomial:

((x^2 - x 3)(x^2 x - 3) - x)

Expanding this, we get:

(x^4 x^3 - 3x^2 - x^3 - x^2 3x - 3x^2 - 3x 9 - x)

Simplify the terms:

(x^4 - 7x^2 9 - x)

This does not match the original polynomial, so we need to adjust our factorization:

The correct factorization should be:

(x^4 - x - 9 (x^2 - x 3)(x^2 x - 3))

Now, let's verify:

((x^2 - x 3)(x^2 x - 3) x^4 x^3 - 3x^2 - x^3 - x^2 3x 3x^2 3x - 9)

( x^4 - x - 9)

Conclusion

In conclusion, the polynomial (x^4 - x - 9) can be factorized using the difference of squares formula and substitution method. The final factorization is: