Polynomial Functions w/ zeros.

[AznBoi]

Ok I have a probelm with find the polynoimal function which has these zeros:

Zeros: 2, 4+sqrt.(5), 4-sqrt.(5)

Find the polynomial equation with the given zeros.

So far I know:

y=(x-2)(x-(4+sqrt.(5))(x-(4-sqrt.(5))

but is there any way I could make the factor (x-(4+sqrt.(5)) into a better one? For example:

Zeros: -2,-1,0,1,2

I did:
y=x(x^2-1)(x^2-4)

instead of y=x(x+2)(x+1)(x-1)(x-2)

Thanks!

 

[HallsofIvy]

Ok I have a probelm with find the polynoimal function which has these zeros:
Zeros: 2, 4+sqrt.(5), 4-sqrt.(5)

Find the polynomial equation with the given zeros.

Not "the" polynomial function- "a" polynomial function. There are an infinite number of polynomial functions having these zeros.

So far I know:

y=(x-2)(x-(4+sqrt.(5))(x-(4-sqrt.(5))

but is there any way I could make the factor (x-(4+sqrt.(5)) into a better one?

Yes, much as you did with -2, 2, and -1, 1 below: (x-(4+\sqrt{5}))(x- (4-\sqrt{5}))= ((x-4)+\sqrt{5})((x-4)-\sqrt{5})= (x-4)^2- 5
= x^2- 8x+ 16- 5= x^2- 8x+ 11
That is the monic polynomial of lowest degree having those roots.

For example:

Zeros: -2,-1,0,1,2

I did:
y=x(x^2-1)(x^2-4)

instead of y=x(x+2)(x+1)(x-1)(x-2)

Thanks!