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Chapter 7: Problem 61

For exercises 1-66, simplify. $$ \frac{2 a^{3}-4 a^{2}-6 a}{4 a^{3}-16 a^{2}-20 a} $$

Short Answer

Expert verified
\( \frac{a - 3}{2(a - 5)} \)

Step by step solution

01

Factor out the greatest common factor from the numerator

The greatest common factor (GCF) of the terms in the numerator is 2a. Factor out 2a from each term: \[ 2a(a^2 - 2a - 3) \]
02

Factor out the greatest common factor from the denominator

The greatest common factor (GCF) of the terms in the denominator is 4a. Factor out 4a from each term:\[ 4a(a^2 - 4a - 5) \]
03

Simplify the fraction

We now have the simplified fraction: \[ \frac{2a(a^2 - 2a - 3)}{4a(a^2 - 4a - 5)} \] We can see that the 2a in the numerator and the 4a in the denominator can be simplified further: \[ \frac{2a}{4a} = \frac{1}{2} \] Thus, the fraction becomes: \[ \frac{a^2 - 2a - 3}{2(a^2 - 4a - 5)} \]
04

Factor the quadratic expressions

To further simplify, factor both quadratic expressions. The numerator factors as follows: \[ a^2 - 2a - 3 = (a - 3)(a + 1) \] The denominator factors as follows: \[ a^2 - 4a - 5 = (a - 5)(a + 1) \]
05

Cancel out common factors

The expression is now: \[ \frac{(a - 3)(a + 1)}{2(a - 5)(a + 1)} \] Cancel out the common factor \(a + 1\) from the numerator and the denominator: \[ \frac{a - 3}{2(a - 5)} \]

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor shared by all terms in an expression. Finding the GCF can greatly simplify algebraic expressions.

To identify the GCF:
  • Break down each term into its prime factors.
  • Identify the common factors across all terms.
  • Multiply these common factors together to find the GCF.

For example, consider the numerator in the exercise, 2a鲁 - 4a虏 - 6a. Each term includes the factor 2a. Hence, 2a is the GCF. The same process applies to the denominator, where 4a is the GCF.
Factoring Quadratics
To simplify quadratic expressions, we factor them into products of binomials. This makes it easier to further reduce the fraction.

For example:
  • Consider the quadratic expression a虏 - 2a - 3.
  • Find two numbers that multiply to the constant term (-3) and add up to the linear coefficient (-2).
  • The numbers -3 and 1 work, hence the factorization is (a - 3)(a + 1).

Similarly, the denominator's quadratic a虏 - 4a - 5 is factored as (a - 5)(a + 1), using the same process.
Simplifying Fractions
Simplifying fractions involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their GCF.

In mathematical terms, if we have a fraction \(\frac{A}{B}\), we find the GCF of A and B, and then rewrite the fraction as \(\frac{A/GCF}{B/GCF}\).

For the given exercise, \(\frac{2a鲁 - 4a虏 - 6a}{4a鲁 - 16a虏 - 20a}\), factor out the GCFs:
  • Numerator's GCF is 2a, resulting in \(\frac{2a(a虏 - 2a - 3)}{4a鲁 - 16a虏 - 20a}\).
  • Denominator's GCF is 4a, leading to \(\frac{2a(a虏 - 2a - 3)}{4a(a虏 - 4a - 5)}\).
  • Simplify by canceling common faction 2a/4a to get \(\frac{1}{2}\).
Canceling Common Factors
Once an algebraic fraction is broken down into its components, check if there's a common factor in both the numerator and the denominator. Canceling these common factors further simplifies the expression.

For example:
  • Original fraction simplified earlier was \(\frac{(a-3)(a+1)}{2(a-5)(a+1)}\).
  • The term (a+1) appears in both the numerator and the denominator and therefore cancels out.

This leaves us with \(\frac{a-3}{2(a-5)}\), which is the most simplified form.

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Most popular questions from this chapter

For exercises \(67-82\), use the five steps and a proportion. Cyclosporine is an anti-rejection drug given to organ transplant patients. A bottle contains \(50 \mathrm{~mL}\) of liquid. Each milliliter of liquid contains \(100 \mathrm{mg}\) of cyclosporine. A kidney transplant patient needs to take \(850 \mathrm{mg}\) of cyclosporine each day. Find the amount of solution that the patient should take each day.

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { and } P \text { are constant; the relationship of } U \text { and } F \text {. } $$

For exercises 43-58, (a) solve. (b) check. $$ \frac{4}{a+6}=\frac{9}{a-4} $$

The relationship of the amount of salad dressing, \(x\), and the amount of sodium in the dressing, \(y\), is a direct variation. Six servings of dressing contain \(1800 \mathrm{mg}\) of sodium. a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the amount of sodium in a bottle that contains 16 servings of salad dressing. d. Use slope-intercept graphing to graph this equation. e. Use the graph to find the amount of sodium in 3 servings of salad dressing.

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { is constant; the relationship of } A \text { and } T \text {. } $$
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