Partial Fraction | Proper Fraction, Improper Fraction, and Equation

To express a single rational fraction as a sum of two or more two rational fractions is called a partial fraction.

Fraction or Rational Fraction

The quotient P(x)/Q(x) of two polynomials P(x) and Q(x) where Q(x) ≠ 0, with no common factor, is called a rational fraction.

Example:

x⁴/x² – 1,x² +1/ x² -1, x+1/x² + 2x – 1 are rational fractions.

There are two types of rational fraction

1. Proper Rational Fraction

2.  Improper Rational Fraction

Proper Rational Fraction

The rational fraction P(x)/Q(x) is called proper fraction if the degree of P(x) is less than degree of Q(x).

Example

x/ x² – 1, x² – 4/x³ + 5x – 3 are proper rational fractions.

When division is impossible then the fraction P(x)/Q(x) is a proper rational fraction.

Improper Rational Fraction

The rational fraction P(x)/Q(x) is called the proper fraction if the degree of P(x) is greater than or equal to the degree of Q(x).

Example

x² / x² – 3, x³– 5/x² +7x – 3, 6x/x + 8 are improper rational fractions.

When division is possible then fraction P(x)/ Q(x) is improper rational fraction.

Equation

An open sentence formed by using the sign of equality “=” is called an equation.

There are two types of the equation.

1. Conditional Equation:

An equation which is satisfied for some particular value of the variable is called conditional equation.

Example

3x = 1 is true only if x = 1/3 whereas x² – 5x +6 = 0 is true for x = 2 and x = 3

2.Identity Equation

 An equation that is satisfied for all values of the variable is called identity.

Example

• (a + b)x = ax + bx is true for all values of x.
• X² – 5x + 6 = (x- 2)(x-3) is true for all values of x.

Mixed Form

A mixed form is the sum of a polynomial and a proper rational fraction.

Example

3x² + 1/ x-2 = 3x + 6 + 13/ x – 2

Partial Fraction

To express a single rational fraction as a sum of two or more two rational fractions is called a partial fraction.

Resolution of Partial fraction by example:

Adding fractions 3/ x- 2 and 2/ x+1

3/ x-2 + 3/ x+1 = 3(x+1) + 2(x-2)/(x-2)(x+1)

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

= 5x – 1/ (x-2)(x+1)

The reverse process to decompose 5x-1/ (x-2)(x+1) into sum of simpler fraction.

3/x-2 and 2/x+1 i.e. 5x-1/(x-2)(x+1)

= 3/x-2 + 2/x+1 is called partial fraction resolution.

Theorem of Partial Fraction

If two polynomials are equal for all values of the variable, then the polynomials have the same degree and the coefficients of like powers of the variable in both the polynomials must be equal.

Example

Px³ + qx² + ax + b = 2x³ – 3x² – 4x + 5 for all x then p = 2, q = -3, a = -4, b = 5

Points for resolution of a rational fraction into a partial fraction

1. Degree of numerator P(x) must be less than the degree of denominator Q(x) if not then divide and work with the remainder theorem.

2. Clear the given equation of fraction.

3. Equate the coefficient of like powers.

4. Solve the resulting equations for the coefficient.

Resolve 2x³ + x² – x – 3/ x(2x + 3)(x-1) into partial equation

Solution

2x³ + x² – x – 3/ x(2x + 3)(x-1)

= 2x³ + x² – x – 3/ x(2x² + x – 3)

= 2x³ + x² – x – 3/ 2x³ + x² – 3x

2x³ + x² – x – 3/ x(x-1)(2x+3) =

1 +2x – 3/ x(x-1)(2x+3)

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