Moderate -0.3 This is a straightforward application of the binomial expansion for negative/fractional powers combined with simple polynomial multiplication. While it requires careful algebraic manipulation and knowledge of the generalized binomial theorem, it's a standard textbook exercise with no conceptual challenges—students simply apply the formula, multiply by (2-x), and collect terms. The mechanics are routine for P3 level, making it slightly easier than average.
2 Expand \(( 2 - x ) ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
State correct unsimplified first two terms of expansion of \((1+2x)^{-\frac{3}{2}}\), e.g. \(1 + (-\frac{3}{2})(2x)\)
B1
State correct unsimplified term in \(x^2\), e.g. \((-\frac{3}{2})(-\frac{3}{2}-1)(2x)^2/2!\)
B1
Obtain sufficient terms of the product of \((2-x)\) and the expansion up to the term in \(x^2\)
M1
Obtain final answer \(2 - 7x + 18x^2\)
A1
Do not ISW
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| State correct unsimplified first two terms of expansion of $(1+2x)^{-\frac{3}{2}}$, e.g. $1 + (-\frac{3}{2})(2x)$ | B1 | |
| State correct unsimplified term in $x^2$, e.g. $(-\frac{3}{2})(-\frac{3}{2}-1)(2x)^2/2!$ | B1 | |
| Obtain sufficient terms of the product of $(2-x)$ and the expansion up to the term in $x^2$ | M1 | |
| Obtain **final** answer $2 - 7x + 18x^2$ | A1 | Do not ISW |
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2 Expand $( 2 - x ) ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2016 Q2 [4]}}