DEPARTMENT OF CIVIL ENGINEERING AND GEOMATICS MATHEMATICS 1 – MTH158S PARTIAL FRACTIONS CHAPTER 68 IN BIRD

Lecturer: A. Cloete

Overview of Partial Fractions

• The concept of partial fractions can be illustrated using simple fractions:
  \frac{1}{4} + \frac{1}{5} = \frac{9}{20}

  • This can be rewritten as:
    \frac{9}{20} = \frac{1}{4} + \frac{1}{5}

  • Here we have resolved \frac{9}{20} into two partial fractions \frac{1}{4} and \frac{1}{5} .

• A similar process is applied to algebraic fractions.

• Define F(x) = \frac{p(x)}{q(x)} where p(x) and q(x) are polynomials in x .

  • F(x) is referred to as a rational fraction.

Division of Rational Fractions

• When the degree of the numerator is greater than or equal to the degree of the denominator:

  1. Perform polynomial long division.

  2. Example:
     \frac{2x^4 + x^2 - 3x + 1}{x^3 - 2x^2 - 3x} = 2x + 4 + \frac{15x^2 + 9x + 1}{x^3 - 2x^2 - 3x}

  • The first two terms can be integrated directly, but the remaining fraction requires further manipulation.

Breaking Down into Partial Fractions

• If the denominator is factorable, the fraction is decomposed into simpler partial fractions.

• There are four distinct cases to consider based on the nature of the factors:

Case 1: Linear Non-Repeated Factors

• Structure: f(x) = (ax+b)(cx+d)(ex+f)

  • Decomposition:
    f(x) = \frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{ex+f}

Case 2: Linear Repeated Factors

• Structure: f(x) = (ax+b)^n

  • Decomposition:
    f(x) = \frac{A}{(ax+b)^n} + \frac{B}{(ax+b)^{n-1}} + \ldots + \frac{P}{ax+b}

Case 3: Quadratic Non-Repeated Factors

• Structure: f(x) = (ax^2+bx+c)(dx^2+ex+f)

  • Decomposition:
    f(x) = \frac{Ax+B}{ax^2+bx+c} + \frac{Cx+D}{dx^2+ex+f}

Case 4: Quadratic Repeated Factors

• Structure: f(x) = (ax^2+bx+c)^n

  • Decomposition:
    f(x) = \frac{Ax+B}{(ax^2+bx+c)^n} + rac{Cx+D}{(ax^2+bx+c)^{n-1}}

Steps for Decomposition

1. Check if the degree of the numerator is greater than or equal to the degree of the denominator.

2. If so, use long division to simplify.

3. Check if the fraction is factorable.

4. If factorable, break it into simpler partial fractions.

   • Preferentially, linear factors are easier to work with, hence always attempt to factor quadratic functions into linear components if possible.

5. Depending on the type of factors, apply the suitable decomposition case.

Examples of Decomposition

Example 1: \frac{x-1}{x^3+x^2-12x}

• Check:

  • Is the degree of (x-1) smaller than the degree of (x^3+x^2-12x)?

  • Yes - No need for long division.

  • Is the denominator factorable?

  • Yes - Factor:
    x^3+x^2-12x = x(x^2+x-12) = x(x+4)(x-3)

• Therefore:
  \frac{x-1}{x(x+4)(x-3)} = \frac{A}{x} + \frac{B}{x+4} + \frac{C}{x-3}

• This leads to:
  x-1 = A(x+4)(x-3) + Bx(x-3) + Cx(x+4)

Solving for Coefficients

• For x = 0 :
  -1 = A(4)(-3) \Rightarrow A = \frac{1}{12}

• For x = -4 :
  -5 = B(-4)(-7) \Rightarrow B = \frac{5}{28}

• For x = 3 :
  2 = C(3)(7) \Rightarrow C = \frac{2}{21}

Result:

\frac{x-1}{x(x+4)(x-3)} = \frac{1}{12}x + \frac{5}{28}(x+4) + \frac{2}{21}(x-3)

Example 2: ( \frac{t^3-t+3}{t^2(t-2)} )

• Here, the degrees are equal, necessitating the use of long division.
  \frac{t^3-t+3}{t^2(t-2)} = 1 + \frac{2t^2-t+3}{t^2(t-2)}

• Now, resolve to find:
  2t^2-t+3 = A(t-2)+B t(t-2)+C t^2

Find Coefficients

• For t = 0 :
  3 = A(-2) \Rightarrow A = -\frac{3}{2}

• For t = 2 :
  9 = 4C \Rightarrow C = \frac{9}{4}

• Substitute and group terms to find B :

Result:

\frac{t^3-t+3}{t^2(t-2)} = 1 - \frac{3}{2}(t-2) - \frac{1}{4}t + \frac{9}{4}(t-2)

Exercises

• Break the following fractions into partial fractions:

1. \frac{3x-7}{(x-1)(x-2)(x-3)}

2. \frac{4x+3}{4x^3+8x^2+3x}

3. \frac{1}{(x+1)(x^2+1)}

4. \frac{1}{x(x+1)^2}

5. \frac{1}{x(x^2+x+1)}

6. \frac{x+1}{x^2(x-1)}

7. \frac{x^3+x^2}{x^2+x-2}

8. \frac{1}{x^4+4x^2+3}

9. \frac{1}{(x+3)(x^2-x-2)}

10. \frac{x^2+x+1}{(x+3)(x-1)(x-2)}

11. \frac{5x^2-1}{x^3-x}

12. \frac{4x^3+2x^2+1}{4x^3-x}

13. \frac{x^2-3}{(x+1)^2(x+2)}

14. \frac{x^2}{(4x^2-1)(2x+3)}

15. \frac{2x^2+1}{(x+2)^3}

16. \frac{x^3}{(x^2+1)^2}

17. \frac{x^2+x}{x^3-x^2+x-1}

18. \frac{2}{(x^3+x^2+4)(x^2+4)}

19. \frac{x^3-3x^2+x-1}{(x^2-4x+4)}

20. \frac{x^4-x^3-3x^2-2x+1}{(x^3+x^2-2x)}