Rational expressions, rational algebraic fractions, partial fractions, expressing a proper algebraic fraction as a sum of partial fractions

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RATIONAL EXPRESSION, RATIONAL ALGEBRAIC FRACTION, PARTIAL FRACTIONS, EXPRESSING A PROPER ALGEBRAIC FRACTION AS A SUM OF PARTIAL FRACTIONS

Definitions.

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Rational expression or function. An algebraic expression which involves no variable in an irreducible radical or under a fractional exponent.

Examples. The expressions 2x² + 1 and 2x + 1/x are rational. So is

However,

are not.

Theorem 1. Any rational expression can be expressed as a quotient g(x) / G(x) of two polynomials.

Rational algebraic fraction. A quotient g(x) / G(x) of two polynomials i.e.a fraction g(x) / G(x) in which the numerator and denominator are both polynomials.

Proper algebraic fraction. A rational algebraic fraction g(x) / G(x) in which the numerator is of lower degree than the denominator.

Examples.

1. x / (x³ + 2)

2. (x² + 5x + 2) / (3x³ - 2x +1)

Improper algebraic fraction. A rational algebraic fraction g(x) / G(x) in which the numerator is not of lower degree than the denominator.

Example. (4x³ + 2x + 1) / (x² + 5x + 2)

● An improper fraction can be expressed as the sum of a polynomial and a proper fraction by dividing the numerator by the denominator.

Fundamental Theorem of Algebra. If G(x) is a polynomial with real coefficients, then G(x) can be written as the product of linear and quadratic factors with real coefficients:

G(x) = c(x - α₁)(x - α₂) ... (x² + b₁x + c₁) (x² + b₂x + c₂) ...

where b_i²- 4c_i < 0.

● Any proper algebraic fraction g(x) / G(x) can thus be written in the form

g(x) / G(x) = g(x) / [ c(x - α₁)(x - α₂) ... (x² + b₁x + c₁) (x² + b₂x + c₂) ... ]

Partial fractions. A set of fractions whose algebraic sum is a given fraction. Any quotient of polynomials for which the numerator is of lesser degree than the denominator can be expressed as a sum of fractions of types

where n is a positive integer and all coefficients are real if all coefficients in the original polynomials were real. Indeed, partial fractions are usually understood to be fractions of these relatively simple types.

The term method of partial fractions is applied to the study of methods of finding these fractions and using them, particularly in integrating certain rational fractions.

James and James. Mathematics Dictionary

Procedure for expressing a proper algebraic fraction g(x) / G(x) as a sum of partial fractions of type

Let us consider the following cases:

Case 1. Distinct linear factors. To each linear factor x - a occurring once in the denominator of a proper fraction, there corresponds a single partial fraction of the form

where A is a constant to be determined. If, for example, the factors of the denominator of the given fraction g(x) / G(x) are all linear and distinct i.e.

G(x) = c(x - a)(x - b) ...

then

Case 2. Repeated linear factors. To each linear factor x - a occurring n times in the denominator of a proper fraction, there corresponds a sum of n partial fractions of the form

where the A’s are constants to be determined. Suppose, for example, that the denominator G(x) can be factored into real linear factors, one or more repeated i.e.

G(x) = c(x - a)(x - b) ...(x - q)^m ...

Then

Case 3. Distinct quadratic factors. To each irreducible quadratic factor x² + bx + c occurring once in the denominator of a proper fraction, there corresponds a single partial fraction of the form

where A and B are constants to be determined. If, for example, the denominator G(x) can be factored into irreducible quadratic factors, all different i.e.

G(x) = c(x² + b₁x + c₁)(x² + b₂x + c₂) ...

then

Case 4. Repeated quadratic factors. To each irreducible quadratic factor x² + bx + c occurring n times in the denominator of a proper fraction, there corresponds a sum of n partial fractions of the form

where the A’s and B’s are constants to be determined. If, for example, the denominator G(x) can be factored into irreducible quadratic factors, one or more repeated i.e.

G(x) = c(x² + b₁x + c₁)(x² + b₂x + c₂) ... (x² + p₁x + q₁)^m ...

then

In summary. If the denominator G(x) can be factored into a mixture of linear and irreducible quadratic factors, some of which may be repeated i.e.

G(x) = c(x - a)(x - b) ...(x - q)^m ...(x² + b₁x + c₁) ... (x² + p₁x + q₁)^m ...

then

General methods for evaluating coefficients. Let

1. Method of undetermined coefficients.. Multiply both sides of equation by D(x) to clear fractions. Then collect like terms, equate the powers of x, and solve the resulting simultaneous equations for the unknown coefficients.

2. Substitution method. Multiply both sides of equation by D(x) to clear fractions. Then let x assume certain convenient values (x = 1, 0, -1, ...) and solve the resulting equations for the unknown coefficients.

Example.

Clearing of fractions,

(1) 6x² - x + 1 = A(x - 1)(x + 1) + Bx(x + 1) + Cx(x - 1)

Substitution method. Let x = 0. Then A = -1. Letting x = 1, B = 3. Letting x = -1, C = 4. Then

Method of undetermined coefficients. Rewriting (1)

6x² - x + 1 = (A + B + C)x² + (B - C)x - A

Equating coefficients of like powers of x

A + B + C = 6

B - C = -1

-A = 1

Solving this system of equations,

A = -1, B = 3, C= 4.

References.

James and James. Mathematics Dictionary

Ayres. Differential and Integral Calculus.

Osgood. Advanced Calculus

Eshbach. Handbook of Engineering Fundamentals.

CRC Standard Mathematical Tables.