Final answer:
To find the partial fraction decomposition of the rational function 12x²-9, we can factor the denominator and use the method of equating coefficients. By solving the resulting equations, we can determine the constants A and B and substitute them back into the partial fraction decomposition.
Step-by-step explanation:
To find the partial fraction decomposition of the rational function 12x²-9, we can factor the denominator. In this case, the denominator is a difference of squares and can be factored as (2x+3)(2x-3). So, the partial fraction decomposition will have two terms, one with a denominator of 2x+3 and the other with a denominator of 2x-3.
We need to find the constants A and B such that:
12x²-9 = A/(2x+3) + B/(2x-3)
To solve for A and B, we can multiply both sides by the common denominator (2x+3)(2x-3). This will give us:
12x²-9 = A(2x-3) + B(2x+3)
Now we can match the coefficients of the like terms on both sides of the equation to get two equations:
12x²-9 = (2A+2B)x + (-3A+3B)
Equating the coefficients, we have:
2A+2B = 12
-3A+3B = -9
Solving these equations will give us the values of A and B, which we can substitute back into the partial fraction decomposition.