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.

6 questions · Standard +0.0

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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.
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.
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.

Show that the binomial expansion of $f(x)$ in ascending powers of $x$ up to and including the term in $x^3$ is $$A - 8x + \frac{x^2}{2} + Bx^3 + ...$$ where $A$ and $B$ are constants to be found. [4]
Use proof by contradiction to prove that the curve with equation $$y = 8 + 8x - \frac{15}{2}x^2$$ does not intersect the curve with equation $$y = A - 8x + \frac{x^2}{2} + Bx^3 \quad 0 < x < \frac{8}{3}$$ where $A$ and $B$ are the constants found in part (a). (Solutions relying on calculator technology are not acceptable.) [4]