- (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of
$$\frac { 1 } { \sqrt { 4 - x } }$$
giving each coefficient in its simplest form.
The expansion can be used to find an approximation to \(\sqrt { 2 }\)
Possible values of \(x\) that could be substituted into this expansion are:
- \(x = - 14\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 18 } } = \frac { \sqrt { 2 } } { 6 }\)
- \(x = 2\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 2 } } = \frac { \sqrt { 2 } } { 2 }\)
- \(x = - \frac { 1 } { 2 }\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { \frac { 9 } { 2 } } } = \frac { \sqrt { 2 } } { 3 }\)
(b) Without evaluating your expansion,
- state, giving a reason, which of the three values of \(x\) should not be used
- state, giving a reason, which of the three values of \(x\) would lead to the most accurate approximation to \(\sqrt { 2 }\)