교과서 질문
Solve each system, using the method indicated.
5x + 2y = -10
3x - 5y = -6 (Gauss-Jordan)
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Ch. 5 - Systems and Matrices
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
6장, 문제 25
Find the partial fraction decomposition for each rational expression. See Examples 1–4. (3x - 2)/((x + 4)(3x2 + 1))
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Identify the form of the partial fraction decomposition. Since the denominator is \((x + 4)(3x^2 + 1)\), where \(x + 4\) is a linear factor and \(3x^2 + 1\) is an irreducible quadratic factor, the decomposition will be of the form: \[ \frac{3x - 2}{(x + 4)(3x^2 + 1)} = \frac{A}{x + 4} + \frac{Bx + C}{3x^2 + 1} \] where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides of the equation by the common denominator \((x + 4)(3x^2 + 1)\) to clear the fractions: \[ 3x - 2 = A(3x^2 + 1) + (Bx + C)(x + 4) \] This step eliminates the denominators and allows us to work with polynomials.
Expand the right-hand side by distributing: \[ A(3x^2 + 1) = 3A x^2 + A \] and \[ (Bx + C)(x + 4) = Bx^2 + 4Bx + Cx + 4C \]. Combine like terms to write the right side as a polynomial in standard form: \[ (3A + B) x^2 + (4B + C) x + (A + 4C) \].
Set up a system of equations by equating the coefficients of corresponding powers of \(x\) from both sides. On the left, the polynomial is \(3x - 2\), which can be written as \(0x^2 + 3x - 2\). So, equate coefficients: \[ \text{Coefficient of } x^2: 0 = 3A + B \] \[ \text{Coefficient of } x: 3 = 4B + C \] \[ \text{Constant term}: -2 = A + 4C \].
Solve the system of equations for \(A\), \(B\), and \(C\). Once these constants are found, substitute them back into the partial fraction form \( \frac{A}{x + 4} + \frac{Bx + C}{3x^2 + 1} \) to complete the decomposition.
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주요 개념
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Partial Fraction Decomposition
Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or solving equations. It involves breaking down the denominator into factors and assigning unknown constants to each fraction.
추천 영상:
4:07
Decomposition of Functions
Factoring and Types of Denominator Factors
Understanding the factorization of the denominator is crucial. Denominators can have linear factors (like x + 4) or irreducible quadratic factors (like 3x^2 + 1). Each type requires a different form in the decomposition: linear factors correspond to constants in the numerator, while quadratic factors require linear expressions.
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가이드 코스
04:36Factor by Grouping
Setting Up and Solving Systems of Equations
After expressing the rational function as a sum of partial fractions with unknown coefficients, you multiply both sides by the denominator to clear fractions. Then, equate coefficients of corresponding powers of x to form a system of linear equations. Solving this system yields the values of the unknown constants.
추천 영상:
가이드 코스
5:48Solving Systems of Equations - Substitution
관련 실천
교과서 질문
Solve each system by elimination. In systems with fractions, first clear denominators.
6x + 7y + 2 = 0
7x - 6y - 26 = 0
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교과서 질문
Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
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교과서 질문
Graph each inequality. x2 + (y + 3)2 ≤ 16
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교과서 질문
Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.
3x + 2y = -9
2x - 5y = -6
791
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교과서 질문
Evaluate each determinant.
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