Binomial expansion

Objective. Expand (a+bx)^n for positive integer n.

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Learn: the key idea

The binomial expansion of (a+bx)ⁿ for positive integer n uses the formula Σ(from k=0 to n) of C(n,k)aⁿ⁻ᵏ(bx)ᵏ, with binomial coefficients C(n,k) found from Pascal's triangle or n!/(k!(n−k)!). Each term's power of x increases while the power of a decreases, and coefficients grow then shrink symmetrically. For (1+2x)³, the expansion is 1+3(2x)+3(2x)²+(2x)³=1+6x+12x²+8x³, found by substituting a=1, b=2, n=3 term by term. Always keep track of both the binomial coefficient and any constants raised to a power within each term.

Worked example

Find the full binomial expansion of (1+x)⁴, giving all terms in ascending powers of x.

1 $Use Pascal's triangle row for n=4: coefficients 1,4,6,4,1$
2 $Attach ascending powers of x to each coefficient$
3 $Write the terms: 1, 4x, 6x², 4x³, x⁴$
4 $Combine into the full expansion$

Answer: 1 + 4x + 6x² + 4x³ + x⁴

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Common mistake. Students forget to raise the constant multiplier to the correct power, expanding (1+3x)² as 1+6x+3x² instead of the correct 1+6x+9x², since the x² term needs 3² not just 3.

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