Write down the first 3 terms in the expansion of (2−x3)6\left(2 - \frac{x}{3}\right)^6, in ascending power of xx.
All terms must be expressed in the simplest form.
Hence, find the term independent of xx in the expansion of (2−x3)6(4x−x)2\left(2 - \frac{x}{3}\right)^6 \left(\frac{4}{x} - x\right)^2.

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Worked solution

4(i)

1. Apply the binomial theorem to expand the first three terms

(2−x3)6=26+(61)25(−x3)+(62)24(−x3)2+…\left(2 - \frac{x}{3}\right)^6 = 2^6 + \binom{6}{1}2^5\left(-\frac{x}{3}\right) + \binom{6}{2}2^4\left(-\frac{x}{3}\right)^2 + \dots

2. Evaluate the coefficients and simplify the terms

=64+6(32)(−x3)+15(16)(x29)+…= 64 + 6(32)\left(-\frac{x}{3}\right) + 15(16)\left(\frac{x^2}{9}\right) + \dots
=64−64x+2409x2+…= 64 - 64x + \frac{240}{9}x^2 + \dots

3. State the final simplified expression for the first three terms

=64−64x+803x2= 64 - 64x + \frac{80}{3}x^2

Answer

64−64x+803x264 - 64x + \frac{80}{3}x^2

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