Final answer:
To find the partial fraction decomposition of (5x+2)/(2x² +7x+6), factor the denominator and write the expression as A/(x+2) + B/(2x+3). Equate the numerators of the fractions and solve the resulting equations to find the values of A and B. Substitute these values back into the partial fraction decomposition.
Step-by-step explanation:
To find the partial fraction decomposition of (5x+2)/(2x² +7x+6), we need to factor the denominator. The denominator can be factored as (x+2)(2x+3). Therefore, the partial fraction decomposition of the given expression is:
(5x+2)/(2x² +7x+6) = A/(x+2) + B/(2x+3)
To determine the values of A and B, we combine the fractions over a common denominator, which is (x+2)(2x+3):
(5x+2)/(2x² +7x+6) = (A(2x+3) + B(x+2))/(x+2)(2x+3)
From here, we equate the numerators of the fractions:
5x+2 = A(2x+3) + B(x+2)
Simplifying and matching the coefficients of the like terms, we get two equations:
5 = 2A + B (equation 1)
2 = 3A + 2B (equation 2)
Solving these two equations simultaneously will give us the values of A and B, which can be substituted back into the partial fraction decomposition.