To expand (a − b)ⁿ, treat it as (a + (−b))ⁿ and keep the minus sign inside brackets. Then (−b)ʳ is negative for odd r and positive for even r, so the terms alternate in sign.
The pattern is the same as in expanding a positive integer power. The only new work is raising a negative number to a power correctly.
How do the signs behave?
For (a − b)ⁿ written in descending powers of a, the sign of the term with bʳ is (−1)ʳ. So the first term is positive, the second negative, the third positive, and so on.
When the power of the second part is even, the sign is positive. When it is odd, the sign is negative. Reading the power off each term tells you its sign without further thought.
A layout for negative parts
- Rewrite the bracket with the minus sign attached to the second part: (2x + (−3))⁴.
- Write the general term with (−3) in brackets: 4Cr (2x)⁴⁻ʳ (−3)ʳ.
- Evaluate (−3)ʳ first: 1, −3, 9, −27, 81.
- Multiply the coefficient, the power of 2x and this signed number.
- Check the pattern: the signs should alternate + − + − +.
Worked example
Expand (2x − 3)⁴.
Step 1, coefficients: 1, 4, 6, 4, 1.
Step 2, terms:
- r = 0: 1 × (2x)⁴ × 1 = 16x⁴
- r = 1: 4 × (2x)³ × (−3) = 4 × 8x³ × (−3) = −96x³
- r = 2: 6 × (2x)² × 9 = 6 × 4x² × 9 = 216x²
- r = 3: 4 × (2x) × (−27) = −216x
- r = 4: 1 × 1 × 81 = 81
Step 3, add:
(2x − 3)⁴ = 16x⁴ − 96x³ + 216x² − 216x + 81
Step 4, check with x = 1: the left side is (2 − 3)⁴ = (−1)⁴ = 1. The right side is 16 − 96 + 216 − 216 + 81 = 1. They agree.
The mistake to watch for
A typical slip is to square the number but keep the minus sign.
Mistaken working: 6 × (2x)² × (−3)² = 6 × 4x² × (−9) = −216x²
The student wrote (−3)² as −9. In fact (−3)² = (−3) × (−3) = 9, so this term should be +216x².
The correction is to evaluate every signed power on its own line before multiplying. An even power of a negative number is always positive. With the wrong term, the check at x = 1 would give −431 instead of 1, which exposes the error.
Check yourself
Try these, then open each answer.
1. Expand (x − 2)⁵.
Show answer
Coefficients 1, 5, 10, 10, 5, 1 and signed powers of −2: 1, −2, 4, −8, 16, −32.
x⁵ − 10x⁴ + 40x³ − 80x² + 80x − 32
Check at x = 1: (−1)⁵ = −1, and 1 − 10 + 40 − 80 + 80 − 32 = −1.
2. Expand (1 − 3x)³.
Show answer
Coefficients 1, 3, 3, 1. Terms: 1, 3 × (−3x) = −9x, 3 × 9x² = 27x², and (−3x)³ = −27x³.
1 − 9x + 27x² − 27x³
Check at x = 1: (−2)³ = −8, and 1 − 9 + 27 − 27 = −8.
3. Find the coefficient of x³ in (2 − x)⁶.
Show answer
General term: 6Cr × 2⁶⁻ʳ × (−x)ʳ. For x³, r = 3. So 6C3 × 2³ × (−1)³ = 20 × 8 × (−1) = −160.
Where this leads next
A quick test is the strongest protection against sign errors, so move on to checking an expansion using a simple substitution. The binomial expansion practice set then mixes signs with every other skill.
Students who lose marks to signs often know the method perfectly. Spotting the pattern in their own working is something our teachers do in online one-to-one Additional Mathematics tuition.