In Exercises 103–114, factor completely. (x+y)⁴−100(x+y)²

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1. Recognize that the given expression \((x + y)^4 - 100(x + y)^2\) can be factored by treating \((x + y)^2\) as a single variable. Let \(u = (x + y)^2\), so the expression becomes \(u^2 - 100u\).
2. Factor out the greatest common factor (GCF) from \(u^2 - 100u\). The GCF is \(u\), so the expression becomes \(u(u - 100)\).
3. Substitute back \((x + y)^2\) for \(u\). This gives \((x + y)^2((x + y)^2 - 100)\).
4. Notice that \((x + y)^2 - 100\) is a difference of squares. Use the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), where \(a = (x + y)\) and \(b = 10\). This factors \((x + y)^2 - 100\) into \((x + y - 10)(x + y + 10)\).
5. Combine all the factors to write the fully factored form of the expression: \((x + y)^2(x + y - 10)(x + y + 10)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. In this case, recognizing the structure of the polynomial allows us to apply factoring techniques effectively.

Difference of Squares

The difference of squares is a specific factoring pattern that states a² - b² = (a - b)(a + b). This concept is crucial for recognizing and simplifying expressions that can be expressed in this form, such as the expression in the given problem, which can be transformed into a difference of squares.

Substitution Method

The substitution method involves replacing a complex expression with a single variable to simplify the factoring process. In this problem, letting u = (x + y)² can simplify the expression, making it easier to factor and solve. This technique is particularly useful for higher-degree polynomials.