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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.