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Chapter 8: Problem 161

In the following exercises, subtract. $$ \frac{z^{2}+6 z}{z^{2}-25}-\frac{3 z+20}{25-z^{2}} $$

Short Answer

Expert verified
\( \frac{z^2 + 9z + 20}{z^2 - 25} \)

Step by step solution

01

Identify the Denominators

The given fractions have denominators \(z^2 - 25\) and \(25 - z^2\). Notice \(25 - z^2 = -(z^2 - 25)\).
02

Rewrite the Denominators

Rewrite the second fraction by changing its denominator to match the first: \(\frac{3z + 20}{25 - z^2} = -\frac{3z + 20}{z^2 - 25}\).
03

Combine the Fractions

With the rewritten denominators, the expression becomes: \(\frac{z^2 + 6z}{z^2 - 25} - \left(-\frac{3z + 20}{z^2 - 25}\right)\).
04

Simplify the Numerator

Combine the numerators of the fractions: \(\frac{z^2 + 6z + 3z + 20}{z^2 - 25}\), which simplifies to \(\frac{z^2 + 9z + 20}{z^2 - 25}\).

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Key Concepts

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

identifying denominators
When subtracting rational expressions, the first step is to identify the denominators of each rational expression. In this exercise, the denominators are: \(z^2 - 25\) and \(25 - z^2\). These denominators look different, but upon closer inspection, you can see they are closely related. The denominator \(25 - z^2\) is the same as \(-(z^2 - 25)\). Recognizing this relationship is crucial for the following steps.
rewriting denominators
Once the denominators are identified, the next step is to rewrite them so they match. This involves changing one of the fractions so that both denominators are identical. Our goal is to have the denominators match without changing the value of the expression. Here, we rewrite the second fraction by factoring the negative sign out of the denominator: \(\frac{3z + 20}{25 - z^2}\) becomes \(-\frac{3z + 20}{z^2 - 25}\). Now, both fractions have the same denominator, \(z^2 - 25\).
combining fractions
With matching denominators, you can now combine the fractions into a single expression. Keep in mind the rules of subtraction for rational expressions: \(fraction鈧� - fraction鈧� = (numerator鈧� - numerator鈧�) / common denominator\). In this problem, we rewrite the expression using the common denominator: \(\frac{z^2 + 6z}{z^2 - 25} - \left(-\frac{3z + 20}{z^2 - 25}\right)\). Note that subtracting a negative is the same as adding the positive: \( -(-\text{{value}}) = + \text{{value}} \). Therefore, we can rewrite it as: \(\frac{z^2 + 6z}{z^2 - 25} + \frac{3z + 20}{z^2 - 25}\).
simplifying expressions
Now, simplify the combined expression. When we add the numerators: \(z^2 + 6z + 3z + 20\), it simplifies to \(z^2 + 9z + 20\). The resulting rational expression is: \(\frac{z^2 + 9z + 20}{z^2 - 25}\). This is the simplified form, with the numerator and denominator fully combined and simplified. Remember to always check if the numerator and denominator can be factored further or canceled out, which might simplify the fraction even more. In this case, \(z^2 + 9z + 20\) and \(z^2 - 25\) don鈥檛 share common factors, so the expression is in its simplest form.

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Most popular questions from this chapter

In the following exercises, solve. The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters. (a) Write the equation that relates the mass to the volume. (b) What is the volume of this liquid if its mass is 128 kilograms?

Write an inverse variation equation to solve the following problems. Paul, a dentist, determined that the number of cavities that develops in his patient's mouth each year varies inversely to the \(\begin{array}{lll}\text { number of } & \text { minutes } & \text { spent }\end{array}\) brushing each night. His patient, Lori, had 4 cavities when brushing her teeth 30 seconds \((0.5\) minutes) each night. (a) Write the equation that relates the number of cavities to the time spent brushing. (b) How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?

In the following exercises, solve. If \(p\) varies inversely with \(q\) and \(p=2\) when \(q=1\) find the equation that relates \(p\) and \(q\).

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