IB — mathsStandard1 min read

The Binomial Theorem

Expand expressions using the binomial theorem for IB Maths. Find specific terms and apply Pascal's triangle.

Tutor AI Team2026-02-25T02:06:13.268Z

The binomial theorem expands $(a+b)^n$ for positive integer $n$ . It's in the IB data booklet.

The Formula

$(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Row $n$ gives the coefficients. Row 4: 1, 4, 6, 4, 1.

The $(r+1)$ th term: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ .

Find the coefficient of $x^3$ in $(2x + 3)^5$ .

$T_{r+1} = \binom{5}{r}(2x)^{5-r}(3)^r$ . For $x^3$ : $5-r = 3$ → $r = 2$ .

$T_3 = \binom{5}{2}(2x)^3(3)^2 = 10 \cdot 8x^3 \cdot 9 = 720x^3$ . Coefficient: 720.

Problem

$(1+x)^6$ : expand first 4 terms. $= 1 + 6x + 15x^2 + 20x^3 + ...$

Solution

Problem

Find the constant term in $(x + \frac{2}{x})^6$ .

$T_{r+1} = \binom{6}{r}x^{6-r}(\frac{2}{x})^r = \binom{6}{r}2^r x^{6-2r}$ .

Constant: $6-2r = 0$ → $r = 3$ . $T_4 = \binom{6}{3}(8) = 160$ .

Solution

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#ib#maths#algebra#binomial

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