1. ## ex.algebra..

Hello!!!
I am new in this forum, and it seems really interesting !
Could you help me with some exercises? I would be grateful !

1) f: R -> R , f(x)=(x-1)(x-2)(x-3)(x-4) , then f(R)=...?
// R=the multitude of real numbers

2) if p is a natural and prime number and A={ k nr. integer, | sqrt(k*k-p*k)
is a natural number } , then the number of elements of A is .....?

3) If M={ m integer nr. | (m*m*m-3*m+2)/(2*m+1) is an integer nr.} , then the minimum and the maximum from M are ....?

Thank you very much!!!!

2. Originally Posted by inzaghina
1) f: R -> R , f(x)=(x-1)(x-2)(x-3)(x-4) , then f(R)=...?
// R=the multitude of real numbers
$(x-1)(x-2)(x-3)(x-4)=[(x-1)(x-4)][(x-2)(x-3)] = (x^2-5x+4)(x^2-5x+6)$
$[(x^2-5x+5)-1][(x^2-5x+5)+1] = (x^2-5x+5)^2 - 1$
For the equation, $(x^2-5x+5)^2 - 1 = y \implies (x^2-5x+5)^2 = y+1$ to have a solution it means, $y+1\geq 0$ then $x^2-5x+5 = \sqrt{y-1}$. Now for this equation to have a solution we require that $25 - 4(5-\sqrt{y-1}) \geq 0$. Solve for $y$.

Can you explain the other prolems?