Binomial Expansion I
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Expand fully using Pascal's triangle or the binomial theorem:
(a)\((x+2)^4\)
(b)\((2x-1)^3\)
(c)\((1+3x)^5\)
(a)
Answer
\(x^4+8x^3+24x^2+32x+16\)
(b)
Answer
\(8x^3-12x^2+6x-1\)
(c)
First 3 terms useful for approximations
Answer
\(1+15x+90x^2+270x^3+405x^4+243x^5\)
Question 2
Find the coefficient of \(x^3\) in the expansion of \((2-3x)^5\).
Term with \(x^3\)
\(inom{5}{3}(2)^2(-3x)^3 = 10 imes4 imes(-27x^3)\)
Question 3
Find the term independent of \(x\) in the expansion of \(\left(x + \dfrac{2}{x}
ight)^6\).
General term
\(inom{6}{r}x^{6-r}\cdot\dfrac{2^r}{x^r}=inom{6}{r}2^r x^{6-2r}\); independent when \(6-2r=0\Rightarrow r=3\)
Substitute \(r=3\)
\(inom{6}{3}\cdot2^3=20 imes8\)
Question 4
The coefficient of \(x^2\) in the expansion of \((1+ax)^6\) is 60. Find \(a\).
Coefficient of \(x^2\)
\(inom{6}{2}a^2=15a^2=60\Rightarrow a^2=4\)
Question 5
Use the binomial expansion of \((1+x)^4\) to find an approximation for \((1.02)^4\). Compare with the exact value.
Expansion
\((1+x)^4=1+4x+6x^2+4x^3+x^4\); substitute \(x=0.02\)
First 3 terms
\(pprox1+0.08+0.0024=1.0824\)
Answer
Approximation: \(1.0824\); Exact: \(1.08243216\)
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