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"Let $f(x)$ be a polynomial such that $f(0) = 4,$ $f(1) = 5,$ and $f(2) = 10.$ Find the remainder when $f(x)$ is divided by $x(x - 1)(x - 2).$"

I'm not so good with doing the diagrams, if I have the graph infront of me I can follow up from there! 
if you've got any hints please share them, thank you. 

6*9*8*7 is not equal to that.  Can you see why that is wrong?

How many numbers between 2000 and 7000 consist of four different digits?

5*9*8*7 = 2520

Why can't you do it?

I'd have to look up those three centres, I always forget them but they are easy enough to find. For me but also for you!

I assume this is what is wanted.

\(y=\frac{2x^2-3}{x^2-4x+1}=2x^2+8x+30+\frac{112x-33}{x^2-4x+1}\\~\\~\\ \displaystyle\lim_{x\rightarrow\pm\infty}y=2x^2+8x+30\)

Here is the graph:

ok, thanks Alan :)

I know your question is a translation.

Unfortunately, it does not make sense.

Managed to do this graphically using a package called Autograph:

Looks like the minimum is 48.

Checked all EVEN powers of 13 up to 13^1000 + 6 and they are all multiples of 7.

سؤالك لا يترجم كسؤال كامل.

thank you so much! really appreciate it!

Gues is also right.

This can be tackled as a quadratic.

\(3( cos x)^2 = 2 cos x + 1\\ let\;y=cosx\\ 3y^2=2y+1\\ 3y^2-2y-1=0\\ 3y^2-3y+1y-1=0\\ 3y(y-1)+1(y-1)=0\\ (3y+1)(y-1)=0\\ y=-1/3\quad or \quad y=1\\ cosx=-1/3 \quad or \quad cosx=1\\ for\quad 0\le x <2\pi\\ x=\pi\pm acos(1/3)\quad or \quad 0\\ x=\pi\pm 1.231 \quad or \quad 0\\ x=1.910,\;\;\;4.373,\;\;\;0 \)

Maximum value found as follows:

It's a quadratic in cos(x).

cos(x) = -1/3, or cos(x) = 1.

thank you so much melody!

Here's a hint for you:

Looks like 7 is a factor of all of them (leap of faith based on the prime factorisation of ten values of n  !!):

(Melody, the value with n = 2 should be 28567 not 28267).

There are 100 pennies.

As follows:

Oops!  You are quite right.  My mistake.  I should have given it more thought instead of jumping to a quick conclusion.

I just looked at the first 2.  

n=1    the expression = 175

n=2    the expression = 28267

The highest common factor is 1

The problem is with your graph.  You have put the 2 in the wrong place,

You have graphed   \(cos(x^2)\)      INSTEAD OF      \((cos\;x)^2\) which is the same as  \(cos^2 x\)

Here is what your graph should look like:

a=1; b=1000!;c= b%(12^a); if(c==0, goto4, goto5);printc, a; a++;if(a<1000, goto2, 0);a=1;b++;if(b<1000, goto2, discard=0;

OUTPUT = n =497 or 12^497

Link to the graph: https://www.desmos.com/calculator/krwd98cjlz

I looked at the points of intersection and noted the x-value.