1.14: Binomial Expansion of a Square Root

Given that

can be expressed in the form

where $\displaystyle p$ and $\displaystyle q$ are constants,

(a) find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(b) Hence expand $\displaystyle \sqrt{9-x}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$, expressing each coefficient as an exact fraction in its lowest terms.

Using the expansion you found in part (b) with a suitable value of $\displaystyle x$,

(c) find an estimate to 5 decimal places for the value of