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Binomial expansion for positive whole n
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The binomial expansion multiplies out (1 + x)n in one go, without expanding bracket after bracket.
The expansion
(1 + x)n = 1 + nx + n(n − 1)2!x2 + n(n − 1)(n − 2)3!x3 + ...
When n is a positive whole number it stops after the xn term.
The coefficients are nCr, which you can find on a calculator or in Pascal's triangle.
When n is a positive whole number it stops after the xn term.
The coefficients are nCr, which you can find on a calculator or in Pascal's triangle.
Brackets with a number
For (1 + 2x)6 replace x by 2x, and keep the brackets:
= 1 + 6(2x) + 15(2x)2 + 20(2x)3 + ...
= 1 + 12x + 60x2 + 160x3 + ...
= 1 + 6(2x) + 15(2x)2 + 20(2x)3 + ...
= 1 + 12x + 60x2 + 160x3 + ...
Approximations
Put a small value of x into the first few terms.
1.026 = (1 + 0.02)6 ≈ 1 + 6(0.02) + 15(0.02)2 = 1.126
1.026 = (1 + 0.02)6 ≈ 1 + 6(0.02) + 15(0.02)2 = 1.126
Find the coefficient of x2 in (1 − 3x)5.
- The x2 term is 5 × 42!(−3x)2
- = 10 × 9x2
Answer: 90
(1 + x)n = 1 + nx + n(n − 1)2!x2 + ... Replace x by the whole term (like 2x or −3x) and keep the brackets.
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