A-level · Beta

Partial fractions

Build a rational function from simpler fractions and compare their graphs.

Change each fraction and trace their sum. Dashed vertical lines mark excluded inputs.

2x−(−1)+1x−(2)=(3)x+(−3)(x−(−1))(x−(2))\frac{2}{x-(-1)}+\frac{1}{x-(2)}=\frac{(3)x+(-3)}{(x-(-1))(x-(2))}
Change each fraction and trace their sum. Dashed vertical lines mark excluded inputs.-4-8-2-4002448xy

The blue curve is the sum of the green and red dashed curves.

Numerator coefficient of x
3
Numerator constant
-3
First fraction at trace
2
Second fraction at trace
-0.5
Sum at trace
1.5
Excluded inputs
-1 and 2

This model uses two distinct linear factors. Excluded inputs remain excluded after cancellation; a zero coefficient can remove an asymptote but leaves a hole in the original domain. The view clips large values near poles.

Predict, test and explain

Step 1 of 3

Predict

Do denominators add when combining two fractions?

InvestigateCombine the two fractions using a common denominator.

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Do denominators add when combining two fractions?

2x+1+1x−2\frac{2}{x+1}+\frac{1}{x-2}

Hint

Keep the original domain restrictions while simplifying.

Reveal answer

Multiply each numerator by the missing factor. Cancellation never restores an excluded input from the original expression.

3x−3(x+1)(x−2)\frac{3x-3}{(x+1)(x-2)}

Build an explanation

Hint 1

Keep the original domain restrictions while simplifying.

Hint 2

Multiply each numerator by the missing factor. Cancellation never restores an excluded input from the original expression.

Hint 3

Use the model to check a prediction, then explain the result.

Worked example

This example uses fixed values, separate from the diagram controls.

Combine 2/(x + 1) + 1/(x − 2).

  1. Use the common denominator (x + 1)(x − 2).
  2. The numerator is 2(x − 2) + (x + 1).
  3. Collect terms and retain x ≠ −1, 2.

Answer: (3x − 3)/[(x + 1)(x − 2)], with x ≠ −1, 2.

Connect this idea

Watch out: Multiply each numerator by the missing factor. Cancellation never restores an excluded input from the original expression.

Specification and learning route

AQA B10 · Edexcel Pure 2.10

A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Factorisation; Algebraic fractions; Identities and equating coefficients; Linear equations.

AQA specification · Edexcel specification

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