2.8.5

Partial Fractions - Practice

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What are Partial Fractions?

If an algebraic fraction has a numerator that is lower in degree than its denominator, we can split it up into partial fractions. This allows us to do calculations more easily.

Illustrative background for What are the linear factors in a denominator?Illustrative background for What are the linear factors in a denominator? ?? "content

What are the linear factors in a denominator?

  • To split up an algebraic fraction into its partial fractions, we need to factorise the denominator into its linear factors.
    • For example, 11x72x33x22x11x7x(2x+1)(x2)\frac{\small 11x-7}{\small 2x^3 -3x^2 -2x} \rightarrow \frac{\small 11x-7}{\small x(2x + 1)(x-2)}
Illustrative background for Fractions with 2 linear factorsIllustrative background for Fractions with 2 linear factors ?? "content

Fractions with 2 linear factors

  • If a fraction has a denominator consisting of 2 different linear factors, we can express it as the sum of partial fractions:
    • For example, 11x7(2x+1)(x2)A2x+1+Bx2\frac{\small 11x-7}{\small (2x + 1)(x-2)} \equiv \frac{\small A}{\small 2x+1} + \frac{\small B}{\small x-2}
  • Where AA and BB are constants.
  • These constants are found by adding the fractions and either substituting in values of xx or by equating coefficients of xx.
Illustrative background for Fractions with 3 linear factorsIllustrative background for Fractions with 3 linear factors ?? "content

Fractions with 3 linear factors

  • If a fraction has a denominator consisting of 3 different linear factors, we can express it as the sum of partial fractions.
    • For example, 2x+4(x+3)(3x+1)(x2)Ax+3+B3x+1+Cx2\frac{\small 2x+4}{\small (x + 3)(3x+1)(x-2)} \equiv \frac{\small A}{\small x+3} + \frac{ \small B}{\small 3x+1}+\frac{\small C}{\small x-2}
  • Where AA, BB and CC are constants to be found by substitution or equating coefficients.
Illustrative background for Fractions with repeated factorsIllustrative background for Fractions with repeated factors ?? "content

Fractions with repeated factors

  • If a fraction has a denominator consisting of 2 different linear factors and one is repeated, we can express it as the sum of partial fractions:
    • For example, 5(x1)(3x+2)2Ax1+B3x+2+C(3x+2)2\frac{5}{(x-1)(3x+2)^2} \equiv \frac{A}{x-1} + \frac{B}{3x+2} + \frac{C}{(3x+2)^2}
  • Where AA, BB and CC are constants to be found by substitution or equating coefficients.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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