Expand \( (1+x)^4 \) fully.
\( 1 + 4x + 6x^2 + 4x^3 + x^4 \).
Binomial Expansion
10 exam-style questions on binomial expansions for positive integer and rational powers. Tap Show answer for full working.
3 Easy4 Medium3 Hard
Find the first three terms in the expansion of \( (1+2x)^5 \) in ascending powers of \( x \).
\( 1 + 10x + 40x^2 + \cdots \).
Find the coefficient of \( x^3 \) in the expansion of \( (2+x)^6 \).
\( \binom{6}{3}2^3 x^3 = 20\cdot8=160 \).
Find the first three terms in the expansion of \( (1-3x)^{-1} \) in ascending powers of \( x \), stating the range of validity.
\( 1 + 3x + 9x^2 + \cdots \), valid for \( |3x|<1 \) i.e. \( |x|<1/3 \).
Find the first three terms in the expansion of \( \sqrt{1+4x} \) in ascending powers of \( x \).
\( (1+4x)^{1/2} = 1 + 2x - 2x^2 + \cdots \) (valid \( |4x|<1 \)).
Expand \( (2+x)^4 \) and hence find the coefficient of \( x^2 \) in \( (2+x)^4(1-x) \).
\( (2+x)^4=16+32x+24x^2+8x^3+x^4 \). Product: coeff of \( x^2 \) is \( 24 - 32 = -8 \).
Find the term independent of \( x \) in the expansion of \( \left(x + \dfrac{2}{x}\right)^6 \).
General term \( \binom{6}{r}x^{6-r}(2/x)^r = \binom{6}{r}2^r x^{6-2r} \). Set \( 6-2r=0 \) ⇒ \( r=3 \). Term \( \binom{6}{3}2^3=20\cdot8=160 \).
Find the first three terms in the expansion of \( (1+2x)^{-3} \) and state the range of values of \( x \) for which the expansion is valid.
\( 1 - 6x + 24x^2 + \cdots \), valid for \( |2x|<1 \) i.e. \( |x|<1/2 \).
Use the binomial expansion to find an approximation for \( \sqrt{1.02} \) correct to 4 decimal places.
\( \sqrt{1.02}=(1+0.02)^{1/2} \approx 1 + \dfrac{1}{2}(0.02) - \dfrac{1}{8}(0.02)^2 = 1 + 0.01 - 0.00005 = 1.00995 \approx 1.0100 \) (4 d.p.).
Find the coefficient of \( x^2 \) in the expansion of \( (1-3x)^{-2}(1+x)^3 \) up to and including the term in \( x^2 \).
\( (1-3x)^{-2}=1+6x+27x^2+\cdots \). \( (1+x)^3=1+3x+3x^2+\cdots \). Product \( x^2 \) coeff: \( 27 + 6\cdot3 + 1\cdot3 = 27+18+3=48 \).