Partial Fractions
Partial Fractions are used to decompose a complex rational expression into two or more simpler fractions. Generally, fractions with algebraic expressions are difficult to solve and hence we use the concepts of partial fractions to split the fractions into numerous subfractions. While decomposition, generally, the denominator is an algebraic expression, and this expression is factorized to facilitate the process of generating partial fractions. A partial fraction is a reverse of the process of the addition of rational expressions.
In the normal process, we perform arithmetic operations across algebraic fractions to obtain a single rational expression. This rational expression, on splitting in the reverse direction involved the process of decomposition of partial fractions and results in the two partial fractions. Let us learn more about partial fractions in the following sections.
1.  What are Partial Fractions? 
2.  General Formulas of Partial Fractions 
3.  Decomposition of Partial Fractions 
4.  Partial Fraction of Improper Fraction 
5.  FAQs on Partial Fractions 
What are Partial Fractions?
When a rational expression is split into the sum of two or more rational expressions, the rational expressions that are a part of the sum are called the partial fractions. This is referred to as splitting the given algebraic fraction into partial fractions. The denominator of the given algebraic expression has to be factorized to obtain the set of partial fractions.
Every factor of the denominator of a rational expression corresponds to a partial fraction. For example, in the above figure, (4x + 1)/[(x + 1)(x  2)] has two factors in the denominator, and hence there are two partial fractions, one with the denominator (x + 1) and the other with the denominator (x  2).
General Formulas of Partial Fractions
In the above example, the numerators of partial fractions are 1 and 3. The numerator of a partial fraction is not always a constant. If the denominator is a linear function, the numerator is constant. And, if the denominator is a quadratic equation, then the numerator is linear. It means, the numerator's degree of a partial fraction is always one less than the denominator's degree. Further, the rational expression needs to be a proper fraction to be decomposed into a partial fraction. Listed below in the table are partial fraction formulas (here, all variables apart from x are constants).
Form of Rational Fraction  Form of Partial Fraction 
(px + q)/(ax + b)  A/(ax + b) 
(px + q)/(ax + b)^{n}  A_{1}/(ax + b) + A_{2}/(ax + b)^{2} + .......... A_{n}/(ax + b)^{n} 
(px^{2} + qx + r)/(ax^{2} + bx + c)  (Ax + B)/(ax^{2} + bx + c) 
(px^{2} + qx + r)/(ax^{2} + bx + c)^{n}  (A_{1}x + B_{1})/(ax^{2} + bx + c) + (A_{2}x + B_{2})/(ax^{2} + bx + c)^{2} + ...(A_{n}x + B_{n})/(ax^{2} + bx + c)^{n} 
Let us look at a few examples of partial fractions.
 4/[(x  1)(x + 5)] = [A/(x  1)] + [B/(x + 5)]
 (3x + 1)/[(2x  1)(x + 2)^{2}] = [A/(2x  1)] + [B/(x + 2)] + [C/(x + 2)^{2}]
 (2x  3)/[(x  2)(x^{2} + 1)] = [A/(x  2)] + [(Bx + C)/(x^{2} + 1)]
In all these examples, A, B, and C are constants to be determined. Let's learn how to find these constants.
Important Notes
The following points would help in gaining a more clear understanding of partial fractions.
 The numerator's degree of a partial fraction is always just 1 less than the denominator's degree.
 When a partial fraction has repeated factors of the form (ax + b)^{n} or (ax^{2 }+ bx + c)^{n}, they correspond to n different partial fractions where the denominators of the partial fractions have exponents 1, 2, 3, ..., n.
 The above partial fractions formulas do not depend upon the numerator of the given rational expression.
 Before applying the above formulas, factorize the denominator as much as possible. Otherwise, the answer won't be accurate
Decomposition of Partial Fractions
As we have seen earlier, the decomposition of partial fractions is writing a rational expression as the sum of two or more partial fractions. The following steps are helpful to understand the process of decomposition of partial fractions.
 Step1: Factorize the numerator and denominator and simplify the rational expression, before doing partial fraction decomposition.
 Step2: Split the rational expression as per the formula for partial fractions. P/((ax + b)^{2} = [A/(ax + b)] + [B/(ax + b)^{2}]. There are different partial fractions formulas based on the numerator and denenominator expression.
 Step3: Take the LCM of the factors of the denominators of the partial fractions, and multiply both sides of the equation with this LCM.
 Step4: Simplify and obtain the values of A and B by comparing coefficients of like terms on both sides.
 Step5: Substitute the values of the constants A and B on the right side of the equation to obtain the partial fraction.
Let us learn this process of partial fractions decomposition (or expansion) by an example.
Example: Find the partial fraction expansion of the expression(4x + 12)/(x^{2 }+ 4x)
Solution:
Always remember to factor the denominator as much as possible before doing the partial fraction decomposition. (4x + 12)/(x^{2 }+ 4x) = (4x + 12)/[x(x + 4)] ; The denominator has nonrepeated linear factors. So, every factor corresponds to a constant in the numerator while writing the partial fractions.
Let us assume that: (4x + 12)/[(x)(x + 4)] = [A/x] + [B/(x + 4)] → (1)
The LCD (Least Common Denominator) of the sum (on the right side) is x(x + 4). Multiplying both sides by x(x + 4), 4x + 12 = A(x + 4) + Bx → (2)
Now we have to solve it for A and B. For that, we set each linear factor to zero.
Substitute x + 4 = 0 , or x = 4 in (2): 4(4) + 12 = A(0) + B(4); 4 = 4B; B = 1.
Substitute x = 0 in (2): 4(0) + 12 = A(0 + 4) + B(0); 12 = 4A; A = 3.
Substitute the values of A and B in (1), we get the partial fractions decomposition of the given expression: (4x + 12)/[x(x + 4)] = [3/x] + [1/(x + 4)]
Partial Fractions of Improper Fraction
When we have to decompose an improper fraction into partial fractions, we first should do the long division. The long division is helpful to give a whole number and a proper fraction. The whole number is the quotient in the long division, and the remainder forms the numerator of the proper fraction, and the denominator is the divisor. The format of the result of the long division would be Quotient + Remainder/Divisor. Let us understand more of this with the help of the below example.
Example: Find the partial fraction decomposition of the expression (x^{3} +4x^{2}  2x  5)/(x^{2}  4x + 4)
Solution: Here, the degree of the numerator (3) is greater than the degree of the denominator (2). So the given fraction is improper. So we have to do the long division first.
Then write the given fraction as Quotient + Remainder/Divisor. Then we get: (x^{3 }+ 4x^{2 } 2x  5)/(x^{2 } 4x + 4) = x + 8 + (26x  37)/(x^{2} 4x + 4).
Here, the fraction on the right side is a proper fraction and hence it can be split into partial fractions. (26x  37)/(x^{2 } 4x + 4) = (26x  37)/(x  2)^{2 }= A/(x  2)+ B/(x  2)^{2}
Now let us try to solve for A and B. Hint: Set each of (x  2) and x one by one to zero to get A and B. You should get A = 26 and B = 15.
Substituting these values in we have: (26x37)/(x^{2}  4x + 4) = [26/(x2)] + [15/(x2)^{2}] Further we have: (x^{3}+ 4x^{2 } 2x  5)/(x^{2} 4x + 4) = x + 8 + [26/(x  2)] + [15/(x  2)^{2 }]
Tips & Tricks
The following tips are helpful to split a fraction into its partial fractions.
 If the denominator has nonrepeated linear factors:
 The constants can be obtained by setting each linear factor to zero.
 If the denominator has either repeated linear factors and/or irreducible quadratic factors:
 Set the linear factors to zero to find the value of some constants.
 Set x = 0 to get at least one another constant.
 Compare the coefficients of x^{3}, x^{2}, ..., etc to find the other constants.
Solved Examples

Example 1: Decompose the following expression into partial fractions.
(x^{4 }+ x^{3 }+ x^{2 }+ 1)/(x^{2 }+ x  2)
Solution:
When we factorize the denominator, we get: x^{2 }+ x  2 = (x + 2)(x  1). The degree of the numerator (4) is greater than that of the denominator (2). So it is an improper fraction. We need to first do the long division.
So the given fraction can be written as: (x^{4 }+ x^{3 }+ x^{2 }+ 1)/(x^{2 }+ x  2) = x^{2 }+ 3 + (3x + 7)/{(x + 2)(x  1)}; Now we will decompose (3x + 7)/{(x + 2)(x  1)} into partial fractions using: (3x + 7)/{(x + 2)(x  1)} = A/(x + 2) + B/(x  1).
Multiplying both sides by the LCD (x + 2)(x  1); 3x + 7 = A(x  1) + B(x + 2).
Substitute x  1 = 0, or x = 1 we have 3 + 7 = 3B ;B = 4/3.
Substitute x + 2 = 0, or x = 2 we have 6 + 7 = 3A ; A= 13/3.
Substitute the values of A and B in the above equation we have:(3x + 7)/{(x + 2)(x  1)} = 13/{3(x + 2)}+ 4/{3(x  1)}. Therefore the partial fractions decomposition of the given expression is: x^{2 }+ 3 13/{3(x + 2)}+ 4/{3(x  1)}

Example 2: Decompose the following rational expression into partial fractions.
(4x^{3 }+ x + 2)/{x^{2}(x^{2 }+ 1)}
Solution:
Look at the denominator. We have x^{2}. It means the linear factor x is repeating. (x^{2 }+ 1) is an irreducible (can't be factorized) quadratic factor. So the given fraction can be decomposed as follows: (4x^{3 }+ x + 2)/{x^{2}(x^{2 }+ 1)} = [A/x] + [B/x^{2}] + [(Cx + D)/(x^{2 }+ 1)];
Multiplying both sides by the LCD x^{2}(x^{2 }+ 1); 4x^{3} + x + 2 = Ax^{3} + Ax + Bx^{2} + B + Cx^{3} + Dx^{2} Setting the linear factor x to 0, i.e., x = 0, we get: 2 = B. Now we do not have any other linear factors to set to zero. So we will expand the righthand side expression. Then we will compare the coefficients of x^{3}, x^{2}, x, and constant.
By comparing the coefficients of x^{3}, we get 4 = A + C.
By comparing the coefficients of x^{2}, we get 0 = B + D.
By comparing the coefficients of x, we get 1 = A. By comparing the constants, we get 2 = B.
By solving these equations, we get: A = 1, B = 2, C = 3, D = 2 . Further, substitute all these values in, the given expression becomes: [1/x] + [2/x^{2}] + [(3x  2)/(x^{2} + 1)]
Therefore we have the resultant partial fraction as (4x^{3 }+ x + 2)/{x^{2}(x^{2 }+ 1)} = [1/x] + [2/x^{2}] + [(3x  2)/(x^{2} + 1)].
FAQs on Partial Fractions
How Do you Solve a Partial Fraction?
The partial fraction is the result of solving a rational expression. First simplify the rational expression by breaking it down into the possible factors for the numerator, and the denominator. Further, split the expression into partial fractions based on the formulas. The formulas for partial fractions depend on the number of factors and the degree of the denominator of the rational expression. Further, find the value of the required constants to solve the partial fractions.
What Is Meant by A Partial Fraction?
The partial is the splitting of the rational expression into two or more fractions. The process of creation of partial fractions is reverse of the process of addition or subtraction of proper fractions. The input for the process of partial fraction is a rational expression, and the result is a set of two or more proper fractions.
What Are the Different Denominator Types In the Partial Fractions?
The different denominator types in a partial fractions is based on the number of factors of the denominator expression, and the degree of the terms in the denominator. The different denominator types of a partial P/(ax + b), P/[(ax + b)(cx + d)], P/(ax + b)^{2}, P/(ax + b)^{3}, P/(ax + b)^{n}.
What Is the Procedure for Partial Fraction Decomposition?
The decomposition of partial fractions is across the following three simple steps.
 Step1: Split the fractions as per the formula for decompostion of partial fractions, and based on the number of denominator terms.
 Step2: Find the LCM of the denominators, and multiply both the
 Step3: Substitute appropriate values to find the values of the constants in the numerators of the partial fractions.
How Do you Know How To Add Partial Fractions?
While writing an expression as the sum of partial fractions, keep the following points in mind:
 The numerator's degree of a partial fraction is always just 1 less than the denominator's degree.
 When a partial fraction has repeated factors of the form (ax+b)^{n} (or) (ax^{2} +bx+c)^{n}, they correspond to n different partial fractions where the denominators of the partial fractions have exponents 1, 2, 3, ..., n.
For more information, go to "What are General Formulas of Partial Fractions?" section of this page. To add to partial fractions, we just make their denominators the same and add.
For example: 3/x + 1/(x + 4) = 3/x . (x + 4)/(x + 4) + 1/(x + 4) .x/x = (3x + 12)/(x^{2} + 4x) + x/(x^{2} + 4x) = (3x + 12 + x)/(x^{2} + 4x)= (4x + 12)/(x^{2} + 4x)
How Do you Solve a Repeated Root Partial Fraction?
When a partial fraction has repeated factors of the form (ax+b)^{n} or (ax^{2}+bx+c)^{n}, they correspond to n different partial fractions where the denominators of the partial fractions have exponents 1, 2, 3, ..., n. For example, if the denominator is of the form (ax+b)^{n}, then the corresponding partial fractions should be of the form A_{1}/(ax + b) + A_{2}/(ax + b)^{2} + .......... A_{n}/(ax + b)^{n}.
How Do you Know When to Use Partial Fractions?
The partial fractions are to be used when the denominator of the fraction is an algebraic expression, and when there is a need to split the fraction. Also the there should be a possibility of getting at least two factors for the algebraic expression in the denominator.
How Many Types of Partial Fractions are there?
The types of partial fractions depend on the number of possible factors of the denominator, and the degree of the factors of the denominator. Broadly there are about three types of partial fractions. The following three types of partial fractions are as follows.
 (px + q)/[(ax + b)(cx + d)] = A/(ax + b) + B/(cx + d)
 (px + q)/[(ax + b)^{2} = A/(ax + b) + B/(ax + b)^{2}
 (px^{2} + qx + r)/[(x + a)(x^{2} + bx + c)] = A/(x + a) + (Bx + c)/(x^{2} + bx + c)
Is Partial Fraction a Proper Fraction?
The partial fraction needs to be a proper fraction. If the given fraction is an improper fraction, the numerator is divided by the denominator to obtain a quotient and a remainder. And the partial fraction in this case would be remainder/denominator.
How Do you Do Partial Fractions with 3 Terms?
The solving of partial fractions with 3 terms is the same as the solving of partial fractions with 2 terms. Further, the two formulas for partial fractions with 3 terms are as follows.
 k/{(x + a)(x + b)(x + c)} = A/(x + a) + B/(x + b) + C/(x + c)
 K/{x(x + a)^{2}) = A/x + B/(x + a) + C/(x + a)^{2}