how to factor x 4 2

2 min read 18-01-2025

Factoring expressions is a fundamental skill in algebra. This article will guide you through factoring the expression x⁴ + 2x² + 1, step-by-step. We'll explore multiple methods and highlight common pitfalls to avoid. Understanding this process will strengthen your algebraic abilities and help you tackle more complex problems.

Understanding the Expression

Before we dive into factoring, let's analyze the expression x⁴ + 2x² + 1. Notice the pattern: it's a trinomial (three terms) where the exponents are even multiples of each other (4, 2, and the implied 0 in the constant term). This suggests a potential perfect square trinomial.

Method 1: Recognizing a Perfect Square Trinomial

This is the most efficient approach. Remember the formula for a perfect square trinomial: (a + b)² = a² + 2ab + b². If we let a = x² and b = 1, we can rewrite our expression as:

(x²)² + 2(x²)(1) + (1)²

This perfectly matches the perfect square trinomial formula. Therefore, the factored form is:

(x² + 1)²

Method 2: Factoring by Grouping (Less Efficient in This Case)

While factoring by grouping is a powerful technique for many expressions, it's less efficient for this specific example. However, let's demonstrate how it could be approached:

Attempting Factoring by Grouping

Rewrite the expression: We can't easily find factors that add up to 2x² and multiply to 1. This method is less straightforward than recognizing the perfect square trinomial.

Why This Method Isn't Ideal Here

Factoring by grouping typically works best when you have four terms that can be grouped into pairs with common factors. Our trinomial only has three terms, making this approach less effective.

Method 3: Using the Quadratic Formula (Less Efficient and Indirect)

Although this method is less efficient, it illustrates a more general approach to factoring certain polynomials. We can use substitution:

Let y = x². Then our expression becomes:

y² + 2y + 1

This is a quadratic equation in terms of 'y'. We can use the quadratic formula:

y = [-b ± √(b² - 4ac)] / 2a

where a = 1, b = 2, and c = 1. Solving this gives y = -1 (a repeated root).

Substituting back x² for y, we have:

x² = -1

This leads to imaginary roots (x = ±i), which isn't directly helpful in factoring the original expression over real numbers. While this highlights the roots, it doesn't give the factored form in a straightforward way for real numbers.

Checking Your Answer

Always verify your answer by expanding the factored form:

(x² + 1)² = (x² + 1)(x² + 1) = x⁴ + x² + x² + 1 = x⁴ + 2x² + 1

This matches our original expression, confirming that (x² + 1)² is the correct factorization.

Conclusion: The Most Efficient Method

The most straightforward and efficient method for factoring x⁴ + 2x² + 1 is to recognize it as a perfect square trinomial, resulting in the factored form (x² + 1)². While other methods can be applied, they are less efficient and may lead to more complex calculations in this specific case. Remember to always check your answer by expanding the factored expression to confirm its accuracy.