In Exercises 39–44, factor by introducing an appropriate substitution.x⁴ − 4x² − 5

Verified step by step guidance

1. Identify a substitution to simplify the expression. Let \( u = x^2 \).
2. Rewrite the original expression in terms of \( u \): \( u^2 - 4u - 5 \).
3. Factor the quadratic expression \( u^2 - 4u - 5 \) by finding two numbers that multiply to -5 and add to -4.
4. The factors of \( u^2 - 4u - 5 \) are \((u - 5)(u + 1)\).
5. Substitute back \( u = x^2 \) to get the factors in terms of \( x \): \((x^2 - 5)(x^2 + 1)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This process is essential for solving polynomial equations and simplifying expressions. In the given expression, recognizing patterns or using techniques like grouping can help identify factors.

Substitution Method

The substitution method is a technique used to simplify complex expressions by replacing a variable with another expression. In this case, substituting x² with a new variable (e.g., y) can transform the quartic polynomial into a quadratic one, making it easier to factor.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. They can be solved using various methods, including factoring, completing the square, or the quadratic formula. Understanding how to factor quadratics is crucial for solving higher-degree polynomials like the one in the question.