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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 14
Capitolo 6, Problema 14

Write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

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First, factor the denominator \(x^{2} + 2x - 15\). To do this, find two numbers that multiply to \(-15\) and add to \(2\).
Rewrite the denominator as the product of its factors: \((x + 5)(x - 3)\).
Set up the partial fraction decomposition for the rational expression \(\frac{9x + 21}{(x + 5)(x - 3)}\) as \(\frac{A}{x + 5} + \frac{B}{x - 3}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides of the equation by the denominator \((x + 5)(x - 3)\) to clear the fractions, resulting in \(9x + 21 = A(x - 3) + B(x + 5)\).
Expand the right side and then equate the coefficients of like terms (coefficients of \(x\) and the constant terms) on both sides to form a system of equations to solve for \(A\) and \(B\).

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Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.
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Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. For example, x² + 2x - 15 factors into (x + 5)(x - 3). Factoring is essential in partial fraction decomposition to identify the denominators of the simpler fractions.
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Solving Quadratic Equations by Factoring

Setting Up and Solving Systems of Equations

After expressing the rational expression as a sum of partial fractions, you equate numerators and solve for unknown coefficients. This process typically involves setting up a system of linear equations by matching coefficients of corresponding powers of x, which is crucial to find the values that complete the decomposition.
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Solving Systems of Equations - Substitution
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