1. ## Binomial expansion help

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 + 2y)^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 + 3x)^8 has a linear approximation of - 2 – 47x.

2. 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 + 2y)^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 + 3x)^8 has a linear approximation of - 2 – 47x.
If you like, I find it easier to use factorials

$(a+b)^n = {n \choose 0}a^nb^0 + {n \choose 1}a^{n-1}b + {n \choose 2}a^{n-2}b^2 + {n \choose 3}a^{n-3}b^3 + ... + {n \choose n}a^0b^n$

Of course ${n \choose 0} = {n \choose n} = b^0 = a^0 = 1$

3. generally...
$(x+y)^n=\sum_{k=0}^{n}\begin{pmatrix}
n\\
k
\end{pmatrix}x^{n-k}y^{k}$

4. for the first question i got the answer $x^4 + (8x^3 * y) + (24x^2 *y^2) +32xy^3 + 16y^4$ can anyone confirm this

5. Hello perryman
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 + 2y)^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 + 3x)^8 has a linear approximation of - 2 – 47x.
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 $(1+3x)^8$, ignoring powers of $x$ above $x^2$, using the formula the others have given you:
$(1+3x)^8 = \binom801^8(3x)^0 + \binom811^7(3x)^1 +\binom821^6(3x)^2 + ...$
Looks scary, doesn't it? But remember that $1^n = 1$ for all values of $n,\; (3x)^0 = 1$ and
$\binom80 = 1$

$\binom81=8$

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

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

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