To find the partial fraction decomposition of the given improper rational expression, which is:
(6x² + 12x - 177) / (x - 30)
1. First, factor the numerator:
6x² + 12x - 177 factors to (2x - 9)(3x + 19)
2. Now, rewrite the expression:
(2x - 9)(3x + 19) / (x - 30)
3. Next, express this fraction as a sum of partial fractions:
A/(x - 30) + B/(2x - 9) + C/(3x + 19)
4. To find the values of A, B, and C, multiply through by the denominator and simplify:
A(2x - 9)(3x + 19) + B(x - 30)(3x + 19) + C(x - 30)(2x - 9) = (2x - 9)(3x + 19)
5. Now, solve for A, B, and C by comparing coefficients of like terms on both sides of the equation.
6. Once you have determined the values of A, B, and C, rewrite the original expression using the values found in the partial fractions.
7. The final result will be the partial fraction decomposition of the improper rational expression.
Remember to carefully follow the steps and calculations to correctly decompose the given expression into partial fractions.
