Factor each polynomial. See Examples 5 and 6. x2−8x+16−$$y2x^2$-8x+16-y^2$$

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 69

Chapter 1, Problem 69

Factor each polynomial. See Examples 5 and 6. $x^{2} - 8 x + 16 - y^{2}$

Verified step by step guidance

Recognize that the polynomial is a difference of two expressions: $x^2 - 8x + 16$ and $y^2$.

Notice that $x^2 - 8x + 16$ is a perfect square trinomial because it can be written as $(x - 4)^2$.

Rewrite the original expression as a difference of squares: $(x - 4)^2 - y^2$.

Apply the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$, where $a = (x - 4)$ and $b = y$.

Write the factored form as $((x - 4) - y)((x - 4) + y)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Quadratic Trinomials

Factoring quadratic trinomials involves expressing a quadratic expression like x² - 8x + 16 as a product of two binomials. This is often done by finding two numbers that multiply to the constant term and add to the coefficient of the linear term. For example, x² - 8x + 16 factors to (x - 4)(x - 4) or (x - 4)².

Difference of Squares

The difference of squares is a special factoring pattern where an expression of the form a² - b² can be factored into (a - b)(a + b). Recognizing this pattern helps simplify expressions quickly. In the given polynomial, after factoring the quadratic part, the expression can be seen as a difference of squares.

Combining Factoring Techniques

Some polynomials require multiple factoring methods applied sequentially. In this problem, first factor the quadratic trinomial, then apply the difference of squares to the resulting expression. Understanding how to combine these techniques is essential for fully factoring complex polynomials.

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