Partial Fraction Decomposition
MA 132 Lecture Note Outlines-Morrison Definitions:
Rational Expression: A polynomial function divided by a polynomial function.
Degree of polynomial: The value of the highest exponent.
Proper Rational Expression: A rational expression where the degree of the numerator is strictly less than the degree of the denominator.
Improper Rational Expression: A rational expression where the degree of the numerator is greater than or equal to the degree of the denominator.
Polynomial Long Division: The process used to convert improper rational expressions into proper rational expressions.
Linear Factor: A factor of the form ax b .
Irreducible Quadratic Factor: A factor of the form ax 2 bx c where the discriminant is negative. b2 4ac 0
Fundamental Theorem of Algebra: All polynomials can be factored into linear and irreducible factors.
Examples of polynomial long division:
4x3 5x 2 3x 7 x 4 2x 2 6x 8
x 2 x 2 3
The process to decompose a fraction:
1. Make sure it is proper. 2. Factor the denominator. 3. For every distinct linear factor, create a fraction with the denominator equal to the factor, then add an unknown constant. 4. For every distinct irreducible quadratic factor, create a fraction with the denominator equal to the factor, then add an unknown linear factor on top. 5. For repeated roots, see examples. 6. Solve. Multiply by the denominator on both sides. Then create a system of equations or substitute in values for x that will make the equations eliminate a unknown.
Examples:
12 x 2 9
x 4 5x 1 dx dx x 2 5x 6 2x 1x 1 2x 3 1 dx dx 2 2 2 x 1 xx 4
Things to remember:
dx 1 x arctan c x2 a2 a a If not proper, use long division first (note proper means smaller power on top) Factor the bottom into linear factors (of the form ax+b) or irreducible quadratics (discriminate ( b2 4ac ) is negative) [Note: FTA says every polynomial reduces to exactly linear factors and irreducible quadratics] Given a rational expression (polynomial divided by a polynomial) o Remember that we are creating proper fractions, so on top of linear factors we use constants, on top of quadratics we use linear terms, no top will ever be greater than a linear term o We must account for any possible combination of bottom terms, so include a fraction for each power (when a factor has a multiplicity greater than one.) 1 Example: x(x 1)(x 2)3 (x2 1)(x2 3x 4)2 A B C D E Fx G Hx I Jx K x (x 1) (x 2) (x 2)2 (x 2)3 (x2 1) (x2 3x 4) (x2 3x 4)2 Solve 9x2 3x 5 (A B C)x2 (A 2B)x (B C) by solving 9 A B C and 3 A 2B and 5 B C simultaneously a b a b When we have a linear over a polynomial, we can separate it into two fractions c c c Mostly we integrate these to ln , use u-substitution, or use the arctan identity.