Final answer:
To find the partial fraction decomposition, factor the denominator and solve a system of equations to find the values of A, B, and C.
Step-by-step explanation:
To find the partial fraction decomposition of (9x^2 + 15x + 42) / (x(x - 7)(x - 6)), first factor the denominator as x(x - 7)(x - 6). The degree of the numerator is 2, which is less than the degree of the denominator (3), so we can use the method of partial fractions. The decomposition will have the form A/x + B/(x - 7) + C/(x - 6). To solve for A, B, and C, you can use the method of equating coefficients. Multiply both sides of the equation by the denominator to eliminate the fractions, distribute, and then match coefficients of like terms on both sides. This will give you a system of linear equations that you can solve to find the values of A, B, and C. Once you have the values, you can substitute them back into the partial fractions decomposition