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Chapter 5: Problem 47

\(\left(y^{2}-81\right) \div(y-9)\)

Short Answer

Expert verified
y + 9

Step by step solution

01

Recognize the Structure

Identify that \(y^2 - 81\) is a difference of squares and can be factored. The difference of squares formula is given by \(a^2 - b^2 = (a-b)(a+b)\).
02

Apply the Difference of Squares

Rewrite \(y^2 - 81\) using the difference of squares formula. Here, \(a = y\) and \(b = 9\). Thus, \(y^2 - 81 = (y - 9)(y + 9)\).
03

Substitute and Simplify

Substitute the factored form into the original expression: \[ \frac{(y-9)(y+9)}{y-9} \].
04

Cancel Common Factors

Cancel out the common term \(y - 9\) in the numerator and the denominator: \[ \frac{(y-9)(y+9)}{y-9} = y + 9 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
In algebra, recognizing the structure of polynomials helps simplify expressions. One common structure is the 'difference of squares'. This means you have two square numbers separated by a minus sign. The 'difference of squares' formula is: \(a^2 - b^2 = (a-b)(a+b)\). In this exercise, the expression \(y^2 - 81\) is a difference of squares. We can rewrite it using the formula by identifying \(a = y\) and \(b = 9\). So, \(y^2 - 81\) can be factored as: \((y-9)(y+9)\) Recognizing and applying the difference of squares is crucial for simplifying polynomial expressions efficiently.
Simplifying Algebraic Expressions
Simplifying algebraic expressions means to reduce them to their simplest form. For this, factorization is an essential tool. By factoring, you can break down complex expressions into simpler components, which makes them easier to handle. In the exercise, after recognizing and applying the difference of squares, we rewrite \(y^2 - 81\) as \((y-9)(y+9)\). Next, substitute this back into the original expression: \[ \frac{(y-9)(y+9)}{y-9} \] This form exposes common factors in the numerator and the denominator, which is the next crucial step.
Canceling Common Factors
In algebra, once an expression is simplified, you often find common factors in the numerator and the denominator. These common factors can be canceled out to further simplify the expression. In the given problem, the expression \[ \frac{(y-9)(y+9)}{y-9} \] has \((y-9)\) as a common factor in both the numerator and the denominator. By canceling out these like terms, the expression reduces to: \[ y + 9 \] This simplifies the solution significantly. Canceling common factors should always be approached carefully to avoid mistakes, ensuring that only like terms which appear in both the numerator and the denominator are eliminated.

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Most popular questions from this chapter

\(\left(x^{2}-12 x+32\right) \div(x-4)\)

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Problem: Simplify: \(\left(6 x^{2}+8 x-3\right) \div(3 x)\) Incorrect Answer: \(\left(6 x^{2}+8 x-3\right) \div(3 x)\) $$ \begin{aligned} &=\frac{6 x^{2}}{3 x}+\frac{8 x}{3 x}-\frac{3}{3 x} \\ &=2 x+\frac{8}{3}-x \\ &=x+\frac{8}{3} \end{aligned} $$

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