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Chapter 3: Problem 4

Simplify each rational expression. $$\frac{x^{2}+2 x-15}{x^{2}-9 x+18}$$

Short Answer

Expert verified
The simplified form of the given rational expression is \(\frac{x+5}{x-6}\).

Step by step solution

01

Factor the Numerator

Firstly, factor the numerator. To factor a quadratic in the form of \(ax^{2}+bx+c\), look for two numbers which multiply to give \(a*c\) and add to give \(b\). Here, in the expression \(x^{2}+2 x-15\), these numbers are 5 and -3, since they multiply to -15 and add up to 2. Therefore, it can be factored as \((x-3)(x+5)\).
02

Factor the Denominator

Follow the same process to factor the denominator as well. In the expression \(x^{2}-9x+18\), the numbers -3 and -6 multiply to give 18 and add up to -9. So it can be factored as \((x-3)(x-6)\).
03

Simplify the Expression

Now that both the numerator and the denominator have been factored, look for a common factor that can be cancelled out. In this case, the term \((x-3)\) appears in both the numerator and the denominator and can therefore be cancelled out. The resultant expression after simplifying by cancelling out the common term is \(\frac{x+5}{x-6}\).

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Key Concepts

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

Factoring Quadratics
Factoring quadratics is a key skill in algebra that helps simplify expressions and solve equations. A quadratic expression is typically in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The goal is to rewrite this quadratic as a product of two binomials. To do this, look for two numbers that multiply to \(a \times c\) and add up to \(b\).

Here's how it works in practice:
  • Identify the quadratic expression: For example, \(x^2 + 2x - 15\).
  • Calculate \(a \times c\): In this case, it is \(1 \times -15 = -15\).
  • Find two numbers that multiply to -15 and sum to 2: These numbers are 5 and -3.
  • Rewrite the expression as \((x - 3)(x + 5)\).
This method helps break down complex quadratic expressions into simpler, more manageable parts.
Simplifying Expressions
Once you have factored both the numerator and the denominator of a rational expression, the next step is to simplify it by cancelling out common factors. This process is crucial because it significantly reduces the complexity of the expression, making it easier to work with or interpret.

To simplify the expression \(\frac{(x-3)(x+5)}{(x-3)(x-6)}\):
  • Identify any common factors in both the numerator and the denominator. Here, \((x-3)\) is a common factor.
  • Cancel the common factor \((x-3)\) from both the numerator and the denominator.
  • Rewrite the simplified expression: \(\frac{x+5}{x-6}\).
This simplification step highlights the elegance of algebraic manipulation by turning complex expressions into simpler, equivalent forms.
Mathematics Problem Solving
Problem solving in mathematics involves a series of logical steps and the application of relevant techniques to arrive at a solution. Tackling rational expressions calls upon both factorization and simplification skills, offering a comprehensive test of algebraic understanding.

Here鈥檚 a strategic approach to solving such problems:
  • Analyze the expression: Understand the structure and identify any obvious patterns or factors.
  • Apply factorization: Break down the components of the expression, both in the numerator and the denominator.
  • Simplify: Look for opportunities to streamline the expression by cancelling out common terms.
  • Verify: Always check your work to ensure all calculations are correct and the simplification is accurate.
Developing a strategy like this not only aids in solving the current problem but also builds foundational skills that are applicable to a wide array of mathematical challenges.

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Most popular questions from this chapter

Give a possible expression for a rational function \(r(x)\) of the following description: the graph of \(r\) is symmetric with respect to the \(y\) -axis; it has a horizontal asymptote \(y=0\) and a vertical asymptote \(x=0,\) with no \(x-\) or \(y^{2}\) intercepts. It may be helpful to sketch the graph of \(r\) first. You may check your answer with a graphing utility.

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=5 x^{6}-7 x^{5}+4 x^{3}-6$$

Solve the rational inequality. $$\frac{x+1}{x^{2}-9}<0$$

To print booklets, it costs \(\$ 300\) plus an additional \(\$ 0.50\) per booklet. What is the average cost per booklet of printing \(x\) booklets? Use this expression to find the average cost per booklet of printing 1000 booklets.

Find a polynomial \(p(x)\) such that \(p(x)>0\) has the solution set \((0,1) \cup(3, \infty) .\) There may be more than one correct answer.
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