Textbook Question
What expression in x represents ?
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Ch. 5 - Systems and Matrices
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 6, Problem 3
Answer each question. By what expression should we multiply each side of (3x - 2)/(x + 4)(3x^2 + 1) = A/(x + 4) + (Bx + C)/(3x^2 + 1) so that there are no fractions in the equation?
Verified step by step guidance1
Identify the denominators on both sides of the equation. The denominators are \((x + 4)\) and \((3x^2 + 1)\).
To eliminate the fractions, multiply each term on both sides of the equation by the least common denominator (LCD) of all denominators present.
The LCD is the product of the distinct factors in the denominators, which is \((x + 4)(3x^2 + 1)\).
Multiply every term on both sides of the equation by this LCD: \((x + 4)(3x^2 + 1)\).
This multiplication will clear all denominators, resulting in an equation without fractions, making it easier to solve for \(A\), \(B\), and \(C\).
Verified video answer for a similar problem:
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Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Fraction Decomposition
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. It involves breaking down a complex fraction into terms with simpler denominators, often linear or quadratic factors, to facilitate integration or solving equations.
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Decomposition of Functions
Common Denominator Multiplication
To eliminate fractions in an equation, multiply both sides by the least common denominator (LCD) of all fractional terms. This clears denominators, resulting in a polynomial equation that is easier to solve or manipulate.
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02:58Rationalizing Denominators
Factoring and Identifying Denominators
Understanding how to factor expressions and identify the denominators involved is crucial. In this problem, recognizing the denominators (x + 4) and (3x^2 + 1) helps determine the LCD, which is their product, to clear fractions effectively.
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02:58Rationalizing Denominators
Related Practice
Textbook Question
What is the value of x if ?
Textbook Question
What is the augmented matrix of the following system?
-3x + 5y = 2
6x + 2y = 7
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Textbook Question
Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 so that there are no fractions in the equation?
Textbook Question
Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3).
2x + y = 1
x2 + y2 = 10
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Textbook Question
Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.
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