Edexcel IAL Sample Assessment Paper 2018 P4 (WMA14/01) Q1 Binomial Expansion
Use the binomial series to find the expansion of
\[\frac{1}{(2 + 5x)^3} \quad |x| < \frac{2}{5}\]
in ascending powers of \( x \), up to and including the term in \( x^3 \).
Give each coefficient as a fraction in its simplest form.
(6)
Solution to Question: Expansion of \(\frac{1}{(2 + 5x)^3}\) up to \( x^3 \).
Key Concepts Used:
- General Binomial Expansion Formula
\[
(1 + X)^n \approx 1 + nX + \frac{n(n-1)}{2!}X^2 + \frac{n(n-1)(n-2)}{3!}X^3 + \dots
\] - Algebraic Preparation: Factor out 2 to achieve the \((1 + x)^n\) form.
Step-by-Step Solution:
-
Prepare the expression
Rewrite the function using a negative index and factor out the constant 2:
\[\frac{1}{(2 + 5x)^3} = (2 + 5x)^{-3} = [2(1 + \frac{5}{2}x)]^{-3} = 2^{-3}(1 + \frac{5}{2}x)^{-3} = \frac{1}{8}(1 + \frac{5}{2}x)^{-3}\] -
Identify \( n \) and \( X \)
\[n = -3 \text{ and } X = \frac{5}{2}x\]
-
Apply the Binomial Formula
\[(1 + \frac{5}{2}x)^{-3} \approx 1 + (-3)\left(\frac{5}{2}x\right) + \frac{(-3)(-4)}{2!}\left(\frac{5}{2}x\right)^2 + \frac{(-3)(-4)(-5)}{3!}\left(\frac{5}{2}x\right)^3 + \dots\]
-
Simplify terms
\[\approx 1 – \frac{15}{2}x + \frac{(12)}{2}\left(\frac{25}{4}x^2\right) + (-20)\left(\frac{125}{8}x^3\right) + \dots\]
\[\approx 1 – \frac{15}{2}x + \frac{150}{4}x^2 – \frac{2500}{8}x^3 + \dots\]
\[\approx 1 – \frac{15}{2}x + \frac{75}{2}x^2 – \frac{625}{4}x^3 + \dots\] -
Multiply by \(\frac{1}{8}\)
\[=\frac{1}{8} \left( 1 – \frac{15}{2}x + \frac{75}{2}x^2 – \frac{625}{4}x^3 + \dots \right)\]
Final Answer:
\[\frac{1}{(2+5x)^3} = \frac{1}{8} – \frac{15}{16}x + \frac{75}{128}x^2 – \frac{625}{256}x^3 + \dots\]