Textbook Question
In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x4−162
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0. Review of Algebra
Factoring Polynomials
2:08 minutesProblem 71
Textbook Question
In Exercises 69–80, factor completely.(x + y)² + 6(x + y) + 9
Verified step by step guidance1
Recognize that the expression \((x + y)^2 + 6(x + y) + 9\) is a quadratic in terms of \(u = (x + y)\).
Rewrite the expression as \(u^2 + 6u + 9\).
Notice that this is a perfect square trinomial, which can be factored as \((u + 3)^2\).
Substitute back \(u = (x + y)\) into the factored form to get \((x + y + 3)^2\).
Verify the factorization by expanding \((x + y + 3)^2\) to ensure it matches the original expression.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting them as a product of binomials. This process is essential for simplifying expressions and solving equations. The standard form of a quadratic is ax² + bx + c, and the goal is to express it in the form (px + q)(rx + s). Recognizing patterns, such as perfect squares or the difference of squares, can aid in this process.
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Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique involves manipulating the expression to create a squared term, which can simplify factoring. For example, in the expression (x + y)² + 6(x + y) + 9, recognizing that it can be rewritten as ((x + y) + 3)² helps in identifying the factors.
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Binomial Theorem
The Binomial Theorem provides a formula for expanding expressions raised to a power, specifically (a + b)ⁿ. In the context of factoring, it helps in recognizing patterns in polynomials. For instance, the expression (x + y)² can be expanded to x² + 2xy + y², which is useful for identifying and factoring quadratic expressions effectively.
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