Binomial Expansion II
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find the first four terms of the expansion of \((1-2x)^7\) in ascending powers of \(x\).
Terms \(r=0,1,2,3\)
\(1+7(-2x)+21(-2x)^2+35(-2x)^3\)
Answer
\(1-14x+84x^2-280x^3+\ldots\)
Question 2
The expansion of \((a+bx)^5\) has first term 32 and second term \(-240x\). Find \(a\) and \(b\).
First term
\(a^5=32\Rightarrow a=2\)
Second term
\(5a^4b=-240\Rightarrow 5\cdot16\cdot b=-240\Rightarrow b=-3\)
Question 3
Find the coefficient of \(x^4\) in the expansion of \((3+x)^3(1-2x)^4\). (Consider only the \(x^4\) producing terms.)
Expand each to relevant terms
\((3+x)^3=27+27x+9x^2+x^3\); \((1-2x)^4=1-8x+24x^2-32x^3+16x^4\)
Collect \(x^4\) contributions
\(27\cdot16+27\cdot(-32)+9\cdot24+1\cdot(-8)=432-864+216-8\)
Question 4
Show that the ratio of the 4th term to the 3rd term in the expansion of \((1+x)^n\) is \(\dfrac{(n-2)x}{3}\).
3rd term (\(r=2\))
\(inom{n}{2}x^2=\dfrac{n(n-1)}{2}x^2\)
4th term (\(r=3\))
\(inom{n}{3}x^3=\dfrac{n(n-1)(n-2)}{6}x^3\)
Ratio
\(\dfrac{n(n-1)(n-2)x^3/6}{n(n-1)x^2/2}=\dfrac{(n-2)x}{3}\) ✓
Question 5
Find the term in \(x^2\) in the expansion of \((2x+1)^3(x-3)^4\).
Relevant terms from each
\((2x+1)^3: 1+6x+12x^2+\ldots\); \((x-3)^4: 81-108x+54x^2-\ldots\)
Collect \(x^2\)
\(1\cdot54+6x\cdot(-108x)+12x^2\cdot81=54-648+972\)
Answer
\(378x^2\); coefficient is \(378\)
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