Express (2x-14)/(x^2+2x-15) as partial fractions
Partial fractions is a method of expressing a single fraction with multiple factors in the denominator as a sum of fractions. First we must factorise the denominator (bottom of the fraction).x2+2x-15 = (x+5)(x-3)We can then begin to write the fraction in partial form: (2x - 14)/[(x+5)(x-3)] = A/(x+5) + B/(x-3)From here, we only need to solve for A and B. First we must multiply through by the denominator.2x - 14 = A(x-3) + B(x+5) There are two methods of continuing. Firstly, and most simply, take values of x that set each of the bracket-coefficients of A and B to zero. We first do this by taking x=3, so we get -8 = 8B => B = -1, then x=-5, -14 = -8A => A = 3. Secondly, we can equate the coefficients of x and x0 to obtain simultaneous equations, then solve them for values of A and B. Equating x, 2 = A + B, then equating x0, -14 = -3A +5B. Solve these as you would any simultaneous equation to achieve the same result. To finish up, we substitute our values of A and B into our partial form giving: (2x - 14)/(x2+2x-15) = 3/(x+5) - 1/(x-3)
