The big idea: A power of a two-term bracket, like , opens out into terms. Each coefficient is — row of Pascal's triangle — times a falling power of the first term and a rising power of the second.
Build each term: Pascal multiplier × falling power of x × rising power of 2. Then simplify.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Expand with the triangle
Expand .
Step by step
- Use row 5. The power is 5, so the Pascal multipliers are . The first part is and the second part is .
- Attach the powers. The power of falls from 5 to 0; the power of 3 rises from 0 to 5. Multiply down each column, just as in the animation.
- Simplify each term. Use , , and .
Final answer
— six terms, with powers of falling from 5 to 0.
Quick check: For this worked example, expect terms. Before simplifying, the powers of and in each term must add to 5.
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Raise the whole term: In , the second part is all of .
Raise both pieces: .
A coefficient inside
Expand .
Step by step
- Row 3 coefficients: . Keep the whole together, with falling powers of 3.
- Raise the whole each time: , .
Final answer
.
A minus changes the signs: In , keep the minus with .
Odd powers are negative; even powers are positive. The signs alternate: .
A minus inside
Expand .
Step by step
- Use as the second term. Row 4 is ; even powers of are , odd powers are .
- Work each out: , , .
Final answer
— the signs alternate because of the minus.
The number one slip: The coefficient and the sign are part of the term: (not ) and (not ).
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You don't always need the whole thing: Often you only need the first few terms, in ascending powers — start at the constant and take with , without drawing the whole triangle.
First four terms
Find the first four terms, in ascending powers of , of .
Step by step
- Take (every power of 1 is still 1, so only is left).
- Work out each coefficient — multiply numbers counting down from 12 on top, over .
- Put them in.
Final answer
Small n or large n: Small power → read it off Pascal's triangle. Large power → use (formula or calculator). Here row 12 would be slow to draw, so wins.
An exam-style question
Expand and simplify .
Step by step
- Row 5 coefficients: . First term , second term .
- Raise the whole each time: , , , .
Final answer
.
To get full marks
- Pair the six row-5 numbers with rising powers of .
- Raise the whole to each power (, , , ), not just the .
- Give every coefficient fully worked — forgetting to raise the 2 loses the simplification marks.