Partial Fractions

Exercises on mastering the art of partial fraction decomposition.

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Express in partial fractions. Your answer must be in a particular format. Click the question number to see the template the answer must match.

 1 $$\dfrac{7x-2}{(x+1)(x-2)}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{x+\square }+\frac{\square }{x-\square }$$ 2 $$\dfrac{7x+13}{(x+3)(x-1)}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{x+\square }+\frac{\square }{x-\square }$$ 3 $$\dfrac{4x-41}{x^2-3x-10}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{x+\square }-\frac{\square }{x-\square }$$ 4 $$\dfrac{x}{2x^2+5x+3}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{\square x+\square }-\frac{\square }{x+\square }$$ 5 $$\dfrac{x}{x^2-5x+4}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{\square (x-\square )}-\frac{\square }{\square (x-\square )}$$ 6 $$\dfrac{3x+2}{x^2-4}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{x+\square }+\frac{\square }{x-\square }$$ 7 $$\dfrac{2x-1}{x^2-2x-3}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square }{\square (x+\square )}+\frac{\square }{\square (x-\square )}$$ 8 $$\dfrac{x+9}{3x^2-5x-2}$$ ☐$$\equiv$$ ☐ ✓ ✗ $$\frac{\square \square }{\square (x-\square )}-\frac{\square \square }{\square (\square x+\square )}$$
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This is Partial Fractions level 1. You can also try:
Level 2 Level 3 Level 4

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Description of Levels

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Before starting this exercise you may want to check out Algebraic Fractions.

Refer to the syllabus for the course you are following to establish which levels are required.

Level 1 - Linear numerator and quadratic denominator decomposing to two fractions with linear denominators

Example:

$$\frac{px+q}{(x-a)(x-b)} \equiv \frac{A}{x-a} + \frac{B}{x-b}$$

Level 2 - Linear or quadratic numerator and cubic denominator decomposing to three fractions with linear denominators

Example:

$$\frac{px^2+qx+r}{(x-a)(x-b)(x-c)} \equiv\frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$$

Level 3 - Linear or quadratic numerator and a denominator with a repeated factor

Examples:

$$\frac{px+q}{(x-a)^2} \equiv \frac{A}{x-a} + \frac{B}{(x-a)^2}$$ $$\frac{px^2+qx+r}{(x-a)^2(x-b)} \equiv \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$$

Level 4 - Linear or quadratic numerator and a denominator with a quadratic factor that cannot be factorised

Example:

$$\frac{px^2+qx+r}{(x-a)(x^2+bx+c)} \equiv \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c}$$

If the degree of the numerator is greater than or equal to the degree of the denominator you will need to divide first. See the exercise on Polynonial Division.

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Example - Level 1

Express as partial fractions $$\frac{3x - 2}{x^2 + x - 12}$$

Solution:

We first factorise the denominator.

$$x^2 + x - 12$$ factorises as $$(x+4)(x-3)$$

Thus, we rewrite the expression in the form:

$$\frac{3x - 2}{(x + 4)(x - 3)} = \frac{A}{x + 4} + \frac{B}{x - 3}$$

Where A and B are constants to be determined.

By multiplying through by the common denominator $$(x+4)(x-3)$$

$$3x - 2 = A(x - 3) + B(x + 4)$$

Expanding and equating coefficients, we obtain:

$$3x - 2 = Ax - 3A + Bx + 4B$$

Combining like terms, this becomes:

$$3x - 2 = (A + B)x + (-3A + 4B)$$

Equating the coefficients of x and the constant terms, we have:

For the coefficients of x: $$3 = A + B$$

For the constant terms: $$-2 = -3A + 4B$$

Solving these equations simultaneously gives:

Let's substitute $$B = 3 - A$$ into the second equation:

$$-2 = -3A + 4(3 - A)$$

This simplifies to:

$$-2 = -3A + 12 - 4A$$

$$-2 = -7A + 12$$

$$-7A = -14$$

$$A = 2$$

Substituting $$A = 2$$ into $$B = 3 - A$$ gives:

$$B = 3 - 2 = 1$$

Therefore, the expression in partial fractions is:

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

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