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Chapter 7: Problem 23

In the following exercises, simplify each rational expression.] $$ \frac{20-5 y}{y^{2}-16} $$

Short Answer

Expert verified
\( \frac{20 - 5y}{y^2 - 16} = \frac{-5}{y + 4} \)

Step by step solution

01

Factor the Numerator

Factor out the greatest common factor (GCF) of the numerator. The numerator is \(20 - 5y\). The GCF is \(5\), so factor it out:\[ 20 - 5y = 5(4 - y) \]
02

Factor the Denominator

Factor the denominator \(y^2 - 16\). Recognize that it is a difference of squares:\[ y^2 - 16 = (y - 4)(y + 4) \]
03

Rewrite the Rational Expression

Substitute the factored forms of the numerator and denominator into the original expression:\[ \frac{20 - 5y}{y^2 - 16} = \frac{5(4 - y)}{(y - 4)(y + 4)} \]
04

Simplify the Fraction

Notice that \(4 - y\) is the same as \(-(y - 4)\). Rewrite the numerator to make cancellation straightforward:\[ \frac{5(4 - y)}{(y - 4)(y + 4)} = \frac{5(-1)(y - 4)}{(y - 4)(y + 4)} \]Now cancel the common factors in the numerator and the denominator:\[ \frac{5(-1)\cancel{(y - 4)}}{\cancel{(y - 4)}(y + 4)} = \frac{-5}{y + 4} \]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a key skill in algebra. It involves rewriting a polynomial as a product of its factors. For example, if you have the expression \(20 - 5y\), you can notice that both terms have a common factor, 5. So, you can factor out the 5:
  • \(20 - 5y = 5(4 - y)\).
This makes the expression easier to work with and simplifies the process of solving equations. Factoring takes practice, but once you understand it, you'll find it to be a very useful tool.
Greatest Common Factor
The Greatest Common Factor (GCF) is the largest number that can evenly divide each term in a polynomial. To find the GCF, you look at the coefficients and variable parts of each term. For instance, in the expression \(20 - 5y\), the numbers 20 and 5 have a GCF of 5. Once you determine the GCF, you can factor it out of each term:
  • \(20 - 5y -> GCF = 5\)
So you get:
  • \(20 - 5y = 5(4 - y)\).
Finding the GCF is essential for simplifying polynomials and making them easier to manage.
Difference of Squares
The difference of squares is a special factoring technique for expressions that fit the form \(a^2 - b^2\). This expression can be factored as follows:
  • \(a^2 - b^2 = (a + b)(a - b)\).
In our example, the denominator \(y^2 - 16\) is a difference of squares because \(16\) is \(4^2\). So, it can be factored as:
  • \(y^2 - 16 = (y + 4)(y - 4)\).
Recognizing and applying the difference of squares formula simplifies expressions and solves polynomial equations more easily.

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