Write the partial fraction decomposition of the given ration

Write the partial fraction decomposition of the given rational expression. x/x^2 + 8x + 7 What is the partial fraction decomposition? x/x^2 + 8x + 7 (Use integers or fractions for any numbers in the expression.)

Solution

We know that, to decompose a fraction, we must start with factoring the denominator.

Here, x² +8x +7 = x² +7x+x +7= x(x+7) +1(x+7) = (x+7)(x+1).Let x/( x² +8x +7) = [a/(x+7) ]+ [b/(x+1)]. Then a(x+1)+b(x+7) = x or, x(a+ b) +(a+7b) = x. Hence a+7b = 0 and a+ b= 1. From the 1^st equation, we get a = -7b. On substituting this value of a in the 2^nd equation, we get -7b+b =1 or, -6b=1so that b=-1/6. Then a = -7b = -7*(-1/6) = 7/6. Hence, x/ (x² +8x +7) = [7/6(x+7)] –[1/6(x+1)]