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Chapter 7: Problem 3

Fill in the blanks. Each fraction on the right side of the equation \(\frac{x-1}{x^{2}-8 x+15}=\frac{-1}{x-3}+\frac{2}{x-5}\) is a=_______.

Short Answer

Expert verified
Each fraction on the right side of the equation is a partial fraction of the fraction on the left side.

Step by step solution

01

Identify polynomial terms

Notice that the denominators \(x-3\) and \(x-5\) on the right side are factors of the denominator \(x^{2}-8x+15\) on the left side.
02

Factoring

Factor the denominator \(x^{2}-8x+15\) on the left side. This can be factored into \((x-3)(x-5)\). So, the left side of the equation can be rewritten as \(\frac{x-1}{(x-3)(x-5)}\).
03

Analyze the Fractions

Given both sides of the equation now, it's clear that each fraction on the right side is a component of the fraction on the left side when the denominator has been factored. Each fraction on the right side is simply the fraction on the left side with one of its factors in the denominator and the corresponding coefficient in the numerator.

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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 core algebra concept that helps break down complex expressions into simpler, manageable parts. Consider the quadratic polynomial \(x^2 - 8x + 15\). To factor this, we need to find two numbers that multiply to the constant term (15) and add up to the linear coefficient (-8). These numbers are -3 and -5.
  • Step 1: Set up the polynomial as \(x^2 - 8x + 15\).
  • Step 2: Look for two numbers that multiply to 15 (the constant) and sum to -8 (the coefficient of \(x\)).
  • Step 3: The numbers are -3 and -5, hence the factors are \((x-3)(x-5)\).
Factoring turns the polynomial from a seemingly daunting expression into a product of binomials, making it easier to work with in solutions like partial fraction decomposition.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are both polynomials. In mathematics, understanding these expressions is crucial as they pop up frequently in algebra and calculus. The given expression \(\frac{x-1}{x^2 - 8x + 15}\) is a rational expression since both the numerator \((x-1)\) and the denominator \((x^2-8x+15)\) are polynomials.Working with rational expressions involves:
  • Simplifying the expression by factoring both the numerator and the denominator when possible.
  • Canceling out common factors if any exist, to make the expression simpler.
  • Understanding that the domain excludes values that make the denominator zero (here, \(x = 3\) and \(x = 5\)).
Rational expressions are foundational in performing operations like addition, subtraction, multiplication, and division of fractions, leading to more complex mathematical procedures.
Algebraic Fractions
Algebraic fractions are a kind of rational expression where both the numerator and the denominator are algebraic expressions. These arise frequently in algebra when solving equations that have variables in the fractions. The given problem \(\frac{x-1}{x^2 - 8x + 15} = \frac{-1}{x-3} + \frac{2}{x-5}\) involves algebraic fractions on both sides.Key things to know about algebraic fractions:
  • The denominator cannot equal zero, as this would make the fraction undefined.
  • Decompose complex fractions into simpler parts, which is the purpose of partial fraction decomposition.
  • To combine algebraic fractions, find a common denominator, which often involves factoring polynomials.
Algebraic fractions require careful manipulation to ensure solutions are correct. They open doors to understanding more advanced topics in algebra and calculus.

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Most popular questions from this chapter

A furniture company produces tables and chairs. Each table requires 1 hour in the assembly center and \(1 \frac{1}{3}\) hours in the finishing center. Each chair requires \(1 \frac{1}{2}\) hours in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. The assembly center is available 12 hours per day, and the finishing center is available 15 hours per day. Find and graph a system of inequalities describing all possible production levels.

Write the partial fraction decomposition of the rational expression. Then assign a value to the constant \(a\) to check the result algebraically and graphically. $$\frac{1}{(x+1)(a-x)}$$

An object moving vertically is at the given heights at the specified times. Find the position equations \(=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\) for the object. At \(t=1\) second, \(s=132\) feet At \(t=2\) seconds, \(s=100\) feet At \(t=3\) seconds, \(s=36\) feet

Network Applying Kirchhoff's Laws to the electrical network in the figure, the currents \(I_{1}, I_{2}\) and \(I_{3}\) are the solution of the system $$\left\\{\begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ 3 I_{1}+2 I_{2} &=7 \\ 2 I_{2}+4 I_{3} &=8 \end{aligned}\right.$$. Find the currents.

Two concentric circles have radii \(x\) and \(y,\) where \(y>x .\) The area between the circles is at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.
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