Find the zeroes algebraically
f(x) = 2x^4 -2x^2 -40
I dont understand the book answer

Question

Find the zeroes algebraically 
f(x) = 2x^4 -2x^2 -40
I dont understand the book answer

perl · Answer

please explain step by step

perl · Answer

the book got + - sqrt 5

campbell_st · Answer

well start by taking 2 as a common factor 

$$f(x) = 2(x^4 - x^2 -20)$$

you have an equation that can be factorised

$$f(x) = 2(x^2 + 4)(x^2 - 5)$$

so you need to solve 

$$x^2 + 4 = 0$$

and 

$$x^2 - 5 = 0$$

perl · Answer

i dont get how you factorised it

perl · Answer

ok let me try this, u = x^2

perl · Answer

so 2 ( u^2 - u-20)

campbell_st · Answer

thats it... so it factors to 

2(u + 4)(u - 5) = 0

anonymous · Answer

Let's first factor out the 2: $f(x)=2(x^4-x^2-20) = 0 \implies x^4-x^2-20=0$. Now let $u=x^2$. So now we get: $u^2-u-20=0$, now solve this quadratic normally for u and after getting the solutions, plug $x^2$ back in for $u$.

@perl

campbell_st · Answer

I love it when people rewrite a previous post to explain things...

anonymous · Answer

@campbell_st I had written 90% of that when you posted yours.

perl · Answer

thankyou genius

perl · Answer

you guys are both great

anonymous · Answer

So stop loving what I wrote because this is my own solution, not copied off yours to show that I know how to do things.

perl · Answer

but wait, my book says the zeroes are sqrt(5) and -sqrt(5) ,

perl · Answer

they ignored the 2i and -2i

campbell_st · Answer

thats correct as the other possible zeros are complex

x^2 = -4... is undefined in the real number system

perl · Answer

so by 'zeroes' they mean real zeroes?

campbell_st · Answer

yes... as you are graphing on a number plane of real numbers...

perl · Answer

hmmm

anonymous · Answer

After solving for u, you get: $(u-5)(u+4)=0 \implies u=5,-4 $ Plug $x^2$ back in for u and solve for x: $x^2=5 \implies (x-5)(x+5)=0 \implies x = \pm 5.$ 
$x^2=-4 \implies x^2+4=0, x \in \emptyset$

$\therefore x = \pm 5$ 

Do you understand now? 

@perl

perl · Answer

then when do we include *all* zeroes, all roots?

anonymous · Answer

Normally, questions only ask for real zeroes. You don't give the complex zeroes unless asked to. @perl

perl · Answer

ok thanks :)))

perl · Answer

y = 4x^3 -20x^2+25x

perl · Answer

find the real zeroes

campbell_st · Answer

well take x out as a common factor and you have 

$$y = x(4x^2 + 20x + 25)$$

the quadratic in brackets is a perfect square... 

so you just need to factor the quadratic for a repeated zero... 

by taking out x as a common factor you have a zero at x = 0

hope this helps