17-22. Give the partial fraction decomposition for the following expressions.
17. (5x - 7) / (x² - 3x + 2)

Verified step by step guidance

1. Start by factoring the denominator of the given rational expression. The denominator is \(x^{2} - 3x + 2\). Find two numbers that multiply to \$2\( and add to \)-3$.
2. Rewrite the denominator as a product of its factors: \(x^{2} - 3x + 2 = (x - 1)(x - 2)\).
3. Set up the partial fraction decomposition form for the expression \(\frac{5x - 7}{(x - 1)(x - 2)}\) as \(\frac{A}{x - 1} + \frac{B}{x - 2}\), where \(A\) and \(B\) are constants to be determined.
4. Multiply both sides of the equation by the denominator \((x - 1)(x - 2)\) to clear the fractions, resulting in \$5x - 7 = A(x - 2) + B(x - 1)$.
5. Expand the right side and collect like terms, then equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations. Solve this system to find the values of \(A\) and \(B\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This method simplifies integration and other 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 - 3x + 2 factors into (x - 1)(x - 2). 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 function as a sum of partial fractions, coefficients are determined by equating numerators and solving the resulting system of linear equations. This step ensures the decomposition accurately represents the original expression.