Tutorial on decomposing complicated fractions into simpler manageable fractions. One of its important applications is in Integration Using Partial Fractions in calculus.
Rules of Decomposition
How to decompose a fraction $\dfrac{P(x)}{Q(x)}$ into partial fractions?
1  Factor completely polynomial $Q(x)$ into factors of the form
Example 1: Decompose into partial fractions
Solution to Example 1:
Multiply both side of the above equation by the least common denominator, $(x  2)(x + 1)$, and simplify to obtain an equation of the form $2 x+5 = A(x + 1) + B(x  2)$ Expand the right side and group like terms $2 x + 5 = x (A + B) + A  2 B$ For the right and left polynomials to be equal we need to have $2 = A + B$ and $5 = A  2 B$ Solve the above system to obtain $A = 3$ and $B = 1$ Substitute $A$ and $B$ in the suggested decomposition above to obtain As an exercise, group terms on the right to obtain the left side
Example 2: Decompose into partial fractions
Solution to Example 2:
Multiply both side of the above equation by $(x + 1)^2$, and simplify to obtain an equation of the form $1  2 x = A(x + 1) + B$ Expand the right side and group like terms $2x + 1 = A x + (A + B)$ For the right and left polynomials to be equal we need to have $ 2 = A$ and $1 = A + B$ Solve the above system to obtain $A =  2$ and $B = 3$ Substitute $A$ and $B$ in the suggested decomposition above to obtain
Example 3: Decompose into partial fractions
Solution to Example 3:
Multiply both side of the above equation by $(x  2)(x^2 + 2 x + 3)$, and simplify to obtain an equation of the form $4 x^2  x + 8 = A(x^2 + 2 x + 3) + (B x + C)(x  2)$ The above equality is true for all values of $x$, let us use $x = 2$ to obtain an equation in $A$ $22 = 11 A$ Solve for $A$ to obtain $A = 2$ In order to find $C$, we use $x = 0$ in the above equality $8 = 6  2 C$ Solve for $C$ to obtain $C = 1$ To find $B$, we now use $x = 1$ in the above equality $11 = 12 + (B  1)(1  2)$ Solve for $B$ to obtain $B = 2$ The given fraction can be decomposed as follows
Exercises: Decompose the following fractions into partial fractions.
