# binomial expansion question - don't know what is being asked?

• January 7th 2011, 05:17 AM
angypangy
binomial expansion question - don't know what is being asked?
Hi

I have this question:
(i) show that (2 + x)^4 = 16 + 32x + 24x^2 + 8x^3 + x^4 for all x.

for all x? What does that mean?

(ii) Find the values of x for which (2 + x)^4 = 16 + 16x + x^4

??

Can someone please explain?

Angus
• January 7th 2011, 05:27 AM
Prove It
i) For all $\displaystyle x$ means it's true no matter what the value of $\displaystyle x$ is Just perform the binomial expansion.

ii) Expand $\displaystyle (2+x)^4$ which you should have done for part i) and then set it equal to $\displaystyle 16+16x+x^4$ and solve.
• January 7th 2011, 05:32 AM
Plato
Quote:

Originally Posted by angypangy
(i) show that (2 + x)^4 = 16 + 32x + 24x^2 + 8x^3 + x^4 for all x.
for all x? What does that mean?

(ii) Find the values of x for which (2 + x)^4 = 16 + 16x + x^4

We know $\left( {a + b} \right)^4 = \sum\limits_{k = 0}^4 {\binom{4}{k}a^{4 - k} b^k }$.
So for the first part let $a=2~\&~b=x.$

For part two solve $16x+24x^2+8x^2=0$.
• January 7th 2011, 05:48 AM
grgrsanjay
i think this is the equation for second one

x(x^2 + 3x +2) = 0

the values of x are 0, -1 ,-2
• January 7th 2011, 08:02 AM
angypangy
Hi

I am still a little confused with this question. I can see how to get factors 0, -1 and -2. But if I use them in (2 + x)^4 - ie 2+ -2 = 0 - 0^4 = 0 - not 16 + 16x + x^4

Can someone please explain?

Angus
• January 7th 2011, 08:18 AM
e^(i*pi)
$(2+x)^4 = x^4+8 x^3+24 x^2+32 x+16$

x^4+8 x^3+24 x^2+32 x+16 = x^4+16x+16[/tex]

$x^4$ and $16$ cancel and when you take 16x from both sides you get

$8x^3+24x^2+16x = 0$

You can factor out $8x$ from this equation to give $8x(x^2+3x+2) = 8x(x+2)(x+1) = 0$

edit: You have to sub in 0, -1 and 2 on the RHS too

For example using 0

$(2+0)^4 = 16+16(0)+(0)^4 \implies 16=16$ and so $x=0$ is a solution
• January 7th 2011, 08:22 AM
Quote:

Originally Posted by angypangy
Hi

I am still a little confused with this question. I can see how to get factors 0, -1 and -2. But if I use them in (2 + x)^4 - ie 2+ -2 = 0 - 0^4 = 0 - not 16 + 16x + x^4

Can someone please explain?

Angus

$(2+x)^4=x^4+8x^3+24x^2+32x+16$

$(2+x)^4=x^4+16x+16\Rightarrow\ \left(x^4+8x^3+24x^2+32x+16\right)-\left(x^4+16x+16\right)=0$

$\Rightarrow\ 8x^3+24x^2+16x=0\Rightarrow\ x^3+3x^2+2x=0$

You can only verify that $x=0,\;-1,\;-2$ cause the equation $x^3+3x^2+2x$ to be zero.
• January 7th 2011, 09:09 AM
angypangy
Ah you sub in val of x on right too. Yes understand now. Thanks.