[QUESTION]

Liveit

Hi guys,
Any chance of a bit of guidance on this?

Let g(x)=ax^2+bx+c, where a,b and c are elements of R.
Given that k is a real number such that g(k)=0
Prove that x-k is a factor of g(x)

Do I substitute in 0 for x?
Should I end up with an answer that is a real number or just letters?
Is g(x) the same as f(x)?
Help on this would be very much appreciated!
Thanks

Liveit

okay I tried it while looking through the book. is it:
g(x)=ax^2+bx+c
g(k)=ak^2+bk+c
=> a(x^2-k^2)+b(x-k)
=>a(x+k)(x-k) +b(x-k)
=>(a+b)(x-k)
x-k is a factor of g(x)-g(k)
g(k)=0
so x-k is a factor of g(x)

I don't know why that is right but I think it is. Is it?

Fremen

Liveit wrote: »

okay I tried it while looking through the book. is it:
g(x)=ax^2+bx+c
g(k)=ak^2+bx+c

g(k)=ak^2+bk+c.
Is this is a typo?

=> a(x^2-k^2)+b(x-k)
=>(x-k)(x^2-xk+k^2) +b(x-k)

x^2 - k^2 doesn't factor like that. You're thinking of x^3-k^3, but you've got the right idea.

I don't know why that is right but I think it is?

Yup, it's right. The trick is, you're just looking at g(x) the whole time. By subtracting off g(k), you're actually subtracting zero, just in a clever way.

Sentences which aren't questions don't end in question marks.
/punctuationnazi. :P

Liveit

Yup that was a typo.

Yup I was wrong about that part. I think I fixed it now.

Yup, it's right. The trick is, you're just looking at g(x) the whole time. By subtracting off g(k), you're actually subtracting zero, just in a clever way.

But why I am I subtracting zero?
I don't know that but thanks anyway for your answer.

Fremen

Well, as it stands you have this function g(x) which you can't do anything with. If you can rewrite it somehow, maybe you'll be able to do something useful with it.

You know that g(k) = 0, i.e. that k is a root of your function.

So you have a useless expression for g(x): ax^2 + bx + c. But you know that

g(x) = g(x) - 0 = g(x) - g(k)

and you can do something useful with your new expression g(x)-g(k), as your own calculation shows.

Liveit

I always like to know what things are used for in real life so....
What kind of things are done with this sort of maths in real life?

Eliot Rosewater

Well I imagine nothings really done with the proof in real life. One just has to be very pernicety, exacting and rigorous in Maths, otherwise the whole system falls apart. If you base a lot of Mathematics on an invalid idea, then all that Mathematics would be near worthless until a proof has come about.

The value of proof in learning is that it gives you an understanding of whats going on beneath these definitions, and it also gives you a bit of practice.


The value of quadratics in real life is huge I imagine. They pop up lots of places. However Im sure an engineer would have better examples than me.

Fremen

Yeah, quadratics come up all over the place. You wouldn't necessarily find a direct physical interpretation of the proof though.
Easiest example of a quadratic in nature is a particle moving under gravity. Ignoring wind resistance, the path of a golf ball looks like the graph of a quadratic equation.