Split Using Partial Fraction Decomposition (x^2)/(x^2+14x+49)

Problem

(x2)/(x2+14*x+49)

Solution

1. Perform polynomial long division because the degree of the numerator is equal to the degree of the denominator.

x2=1*(x2+14*x+49)−14*x−49

(x2)/(x2+14*x+49)=1−(14*x+49)/(x2+14*x+49)

2. Factor the denominator of the remaining fraction, which is a perfect square trinomial.

x2+14*x+49=(x+7)2

3. Set up the partial fraction decomposition for the fractional part using the repeated linear factor (x+7)2

(14*x+49)/((x+7)2)=A/(x+7)+B/((x+7)2)

4. Solve for the constants by multiplying both sides by the common denominator (x+7)2

14*x+49=A*(x+7)+B

14*x+49=A*x+7*A+B

5. Equate the coefficients of like terms to find A and B

A=14

7*A+B=49

7*(14)+B=49

98+B=49

B=−49

6. Substitute the constants back into the decomposition and combine with the integer part from step 1.

(x2)/((x+7)2)=1−(14/(x+7)−49/((x+7)2))

(x2)/((x+7)2)=1−14/(x+7)+49/((x+7)2)

Final Answer

(x2)/(x2+14*x+49)=1−14/(x+7)+49/((x+7)2)

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