Express as partial fractions

Question

I found the questions in one of the math textbooks. Tried to solve them, however couldn't find the answer.

 

The question "Express the following fractions as partial fractions:"

 

1) (x+1)/(x²-6x+9)

 

2) (x-2)/(x²+4x+4)(x²-4)

 

Please help me with my problem

 

Your sincerly,

A Student

 

Answer 1

Hello Mewnid,

 

To express the following fractions as partial fractions means to split the single fraction up to find the "parts" that make it.

 

1.) (x+1)/(x²-6x+9)

 

Partial fraction decomposition starts with factoring the denominator.

 

(x+1)/(x-3)²

 Write a partial fraction for each factor. * In the special case where you have a perfect square for a denominator.  Write one of the partial fractions over the whole square.

 

(x+1)/(x-3)² = A/(x-3)² + B/(x-3)

 

multiply both sides by (x-3)² so we no longer have fractions. This step is important for determining the correct numerator for each partial fraction. 

 

(x+1) = A + B(x-3)

Pick a value for x that eliminates one variable and solve.

x = 3

4 = A

Pick another value for x, set A = 4 and solve for B.

x =1

2 = 4 -2B

-2 = -2B

1 = B

 

So now we have our numerators. Plug them into the original fractions.

 

(x+1)/(x-3)² = A/(x-3)² + B/(x-3) = 4/(x-3)² + 1/(x-3)

 

The second problem is a higher order quadratic function.  You will need a partial fraction for partial fraction for each exponent from above 1.

 
2.) (x-2)/(x²+4x+4)(x²- 4)

 

Factor the denominator.

 

(x+2)(x+2)(x²- 4)

(x+2)²(x+2)(x-2)  = (x+2)³(x-2)

 

** One of the factors, (x-2) cancels out.

leaving us with;

 

1/(x+2)³

 

This is the partial fraction of (x-2)/(x²+4x+4)(x²- 4) = 1/(x+2)³ 

Answer 2

This problem can be simplified by simple algebra.

 

x²+4x+4=(x+2)^2

 

x²-4=(x+2)(x-2)

 

S othe denominator is=(x-2)*(x+2)³ 

 

Numerator also  has x-2 which cancel each other.

 

So the function is 1/(x+2)³ which is in partial fraction form already.

Thank you