A-level · Beta
Binomial expansion
Explore Pascal’s triangle, exact coefficients and rational-power approximations.
Choose a coefficient in Pascal’s triangle. Drag X to compare the exact value with its expansion.
Tab to a handle, then use the arrow keys. Use the sliders for precise values.
The finite expansion agrees with the original expression for every x.
- Selected term
- Selected coefficient
- Exact function value
- Expansion value
- Absolute error
Each coefficient counts how many ways you can choose the x term from that many brackets.
Predict, test and explain
Step 1 of 3
Predict
What is the coefficient of x² in (1 + x)⁵?
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.
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Hints, self-check and connections
Try a self-check
Fixed values, separate from the diagram controls.
What is the coefficient of x² in (1 + x)⁵?
Hint
Row n gives the coefficients of (1 + x)ⁿ.
Reveal answer
There are 5 × 4 ÷ 2 = 10 pairs of brackets from which to select x.
Build an explanation
Hint 1
Row n gives the coefficients of (1 + x)ⁿ.
Hint 2
The coefficient of xᵏ is n choose k.
Hint 3
For a non-integer power, a finite number of terms is an approximation.
Worked example
This example uses fixed values, separate from the diagram controls.
Expand (1 + x)⁴.
- Read row 4 of Pascal’s triangle: 1, 4, 6, 4, 1.
- Pair the coefficients with powers 0, 1, 2, 3 and 4.
- The integer-power expansion ends at x⁴.
Answer: 1 + 4x + 6x² + 4x³ + x⁴.
Connect this idea
Watch out: The exponent alone is not every coefficient. For x², count pairs of brackets.
Specification and learning route
AQA D1 · Edexcel Pure 4.1
AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.
Useful starting knowledge: Expanding brackets; Indices; Factorials and combinations.