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Chapter 7: Problem 52

$$ \frac{1}{x-1}-\frac{4}{x-2}+\frac{4}{x-3}-\frac{1}{x-4}=\frac{1}{30} $$

Short Answer

Expert verified
The short answer based on the given step-by-step solution would be: Find the common denominator, (x-1)(x-2)(x-3)(x-4), and rewrite the equation with the common denominator. Combine the numerators and cross-multiply to eliminate the fractions. Expand the terms, simplify the equation, and solve the polynomial equation for x. Check the solutions obtained in the original equation to ensure that they do not cause any of the denominators to become zero.

Step by step solution

01

Find the common denominator

To find the common denominator, we need to look at the denominators of each of the given fractions. They are (x-1), (x-2), (x-3), and (x-4). The common denominator will be the product of these four expressions: Common denominator = (x-1)(x-2)(x-3)(x-4)
02

Rewrite the equation with the common denominator

Now we rewrite each term in the equation with the common denominator: \[ \frac{(x-2)(x-3)(x-4)}{(x-1)(x-2)(x-3)(x-4)} - \frac{4(x-1)(x-3)(x-4)}{(x-1)(x-2)(x-3)(x-4)} \\ + \frac{4(x-1)(x-2)(x-4)}{(x-1)(x-2)(x-3)(x-4)} - \frac{(x-1)(x-2)(x-3)}{(x-1)(x-2)(x-3)(x-4)}=\frac{1}{30} \]
03

Simplify the equation

Now that all the terms have the same denominator, we can combine the numerators: \[ \frac{(x-2)(x-3)(x-4) - 4(x-1)(x-3)(x-4) + 4(x-1)(x-2)(x-4) - (x-1)(x-2)(x-3)}{(x-1)(x-2)(x-3)(x-4)}=\frac{1}{30} \]
04

Cross-multiply to eliminate the fractions

Now we will cross-multiply to eliminate the fractions: 30 \(([(x-2)(x-3)(x-4) - 4(x-1)(x-3)(x-4) + 4(x-1)(x-2)(x-4) - (x-1)(x-2)(x-3)]\) =\((x-1)(x-2)(x-3)(x-4)\)
05

Expand the terms and simplify

We will expand all the terms and combine the like terms: 30 [(x虏 - 5x + 6)(x - 4) - 4(x虏 - 4x + 3)(x - 4) + 4(x虏 - 3x + 2)(x - 4) - (x虏 - 3x + 2)(x - 3)] = (x鈦� - 10x鲁 + 35x虏 - 50x + 24) Upon expanding and combining like terms, we will get a polynomial equation in x.
06

Solve the polynomial equation

Now we need to solve the polynomial equation for x. This can be done using a method such as factoring, synthetic division, or by finding the roots using the Rational Root Theorem and Descartes' Rule of Signs. By solving the polynomial equation, we can find the possible values for x that satisfy the given equation. Note: It is essential to check the solutions you obtain in the original equation to ensure that they do not cause any of the denominators to become zero, which would make the original equation undefined.

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Key Concepts

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

Common Denominator
When dealing with fractions, finding a common denominator is crucial for adding or subtracting them. In this exercise, our goal is to combine fractions with different denominators: \((x-1)\), \((x-2)\), \((x-3)\), and \((x-4)\). A common denominator is simply a shared base that allows us to rewrite each fraction equally.

To find the common denominator, multiply the unique factors from each denominator. Here, it's a product of all individual parts:
  • \((x-1)(x-2)(x-3)(x-4)\)
This ensures that each fraction can be expressed with the same denominator, making it easier to combine them in later steps.
Fraction Simplification
Simplifying fractions means rewriting them in a form where both the numerator and denominator are easier to work with. After finding a common denominator, we rewrite each term in the given problem over this common base.

Each term must be adjusted:
  • Multiply both the numerator and denominator by necessary coefficients to match the common denominator.
For example, the term \(\frac{1}{x-1}\) becomes \((x-2)(x-3)(x-4)\) over the common denominator. This adjustment occurs so that all fractions have the same denominator without altering their value.
Cross Multiplication
Cross multiplication is a method used to eliminate fractions by multiplying across the equation. This technique helps to solve equations involving fractions more directly.

Once all terms in a fraction equation share a common denominator, cross-multiplying simplifies the process. For the given exercise:
  • Both sides of the equation are multiplied by the common denominator \((x-1)(x-2)(x-3)(x-4)\)
This eliminates the fraction and helps convert the equation into a polynomial form, making it easier to solve for the variable.
Polynomial Roots
Finding the roots of a polynomial is the process of solving an equation to find values that satisfy it. After cross-multiplying in our exercise, we end up with a polynomial equation.

To solve it, use methods such as:
  • Factoring: Breaking down the polynomial into simpler components.
  • Synthetic Division: A streamlined way to divide polynomials.
  • Rational Root Theorem: Identifies possible rational solutions.
Remember to check each solution against the original equation. Make sure none of the solutions make any denominator zero, as this would make the equation undefined. This final verification ensures that all solutions are valid.

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Most popular questions from this chapter

$$ x^{4}-12 x^{3}+54 x^{2}-108 x+81 $$

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Find the nature of roots of the polynomial \(P(x)=2 x^{8}+3 x^{4}+x^{2}+7\).
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