If an equation has multiple roots of same value , What does it signify?

Question

$(x-1)^{560}$ is said to have 560 roots. Now , all the roots must be 1(since it is the only factor tha can be formed ). And another example is $x^2 - 2x + 1 = 0 $ , Here $ x = 1$ ,only root as answer.

Q1 By saying $560$ roots , It does not give an difference to the answer( I mean that there is only root possible ) but does it give a difference in the graph of the function ?.

Answer 1

The roots of these functions

$$f(x)=(x-1)^{560} \\ g(x)=x^2-4x+4=(x-2)^2$$

are not equal. This means that the graphs are also different.

I accept

$$f(x)=(x-1)^{560}, ~g(x)=(x-1)^2$$

Both functions get equal value at the point $x = 0,1,2.$ This means that the graphs pass through the same point, but it does not mean the graphs are the "same".

Just notice that:

• If $x≠0,1,2$, then $$(x-1)^2≠(x-1)^{560}.$$

This means, graps can never be the "same".

When will the graph of two functions be the same?

Of course, when these functions are equivalent functions.

For example:

$$\color{blue}{f(x)=|x|} \qquad \color{red}{g(x)=\sqrt {x^2}}$$

Since the values ​​of the functions are the same at all points, the graphs are also the same.

Answer 2

Roots of multiplicity become interesting when looking at the structure of a polynomial.

But, if you only care about the graph... if the multiplicity is even, the graph will touch the x-axis but not "cut through" the x-axis. If it is odd, it will cut through.

The graph will be "flat" in the neighborhood of a root of high multiplicity.

The graph of $y = (x-1)^{560}$ will be very nearly zero for all x in (0,2), and then quickly shoot up toward infinity for any x outside of this range.