Form (a+bx)^n requiring factorisation

Questions where the expression is in the form (a+bx)^n with a≠1, requiring factorisation to (a^n)(1+bx/a)^n before applying the binomial theorem.

2 questions

Edexcel C4 2008 January Q2
2. (a) Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 8 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each term as a simplified fraction.
(b) Use your expansion, with a suitable value of \(x\), to obtain an approximation to \(\sqrt [ 3 ] { } ( 7.7 )\). Give your answer to 7 decimal places.
Edexcel Paper 2 2023 June Q13
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of
    $$( 3 + x ) ^ { - 2 }$$ writing each term in simplest form.
  2. Using the answer to part (a) and using algebraic integration, estimate the value of $$\int _ { 0.2 } ^ { 0.4 } \frac { 6 x } { ( 3 + x ) ^ { 2 } } d x$$ giving your answer to 4 significant figures.
  3. Find, using algebraic integration, the exact value of $$\int _ { 0.2 } ^ { 0.4 } \frac { 6 x } { ( 3 + x ) ^ { 2 } } d x$$ giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are constants to be found.