Hello perryman Quote:

Originally Posted by

**perryman** i have two questions i need help with.

for the first one should i use pascals triangle?

1. Expand, fully, each of the following

i) (*x *+ 2*y*)^4

2. Show that, if *x* is small enough for *x*^2 and higher powers of *x *to be neglected, the function (*x* – 2)(1 + 3*x*)^8 has a linear approximation of - 2 – 47*x*.

**e^(i*pi)** and **Raoh** have given you the formula you need to expand #1. Do you understand how to use it?

To give you an example, and to help you with #2, I'll expand $\displaystyle (1+3x)^8$, ignoring powers of $\displaystyle x$ above $\displaystyle x^2$, using the formula the others have given you: $\displaystyle (1+3x)^8 = \binom801^8(3x)^0 + \binom811^7(3x)^1 +\binom821^6(3x)^2 + ...$

Looks scary, doesn't it? But remember that $\displaystyle 1^n = 1$ for all values of $\displaystyle n,\; (3x)^0 = 1$ and$\displaystyle \binom80 = 1$

$\displaystyle \binom81=8$

$\displaystyle \binom82 = \frac{8.7}{2!}$

So, it's simply:$\displaystyle 1 +8(3x) + \frac{8.7}{2!}(3x)^2 + ...$$\displaystyle =1+24x+...$ (and I've left the $\displaystyle x^2$ term out now, because we don't need it anyway - I just put it there as an example)

So if we ignore $\displaystyle x^2$ and higher powers of $\displaystyle x$:$\displaystyle (x-2)(1+3x)^8 = (x-2)(1+24 x + ...)$

Can you complete it now?

Grandad

PS I've just read your answer to #1. Good work!