how to know spot a repeated root in a graph

Question

The diagram shows a cubic curve passing through (–1, 0), (2, 0) and (0, –8).

What is the equation of the curve.

So I would have said:

$-8 = k(x + 1)(x-2)$

$k = 4$

$y = 4(x+1)(x-2)$ but the root at 2 is repeated, how would I have been able to spot this?

Answer 1

Because there is no sign change at $x=2$.

At a simple root (or more generally a root of odd order), a polynomial changes its sign. Indeed, if $a$ is a root of $f$, then we can write $f(x)=(x-a)g(x)$, and as $x-a$ changes its sign at $x=a$, we either have that $f$ changes its sign and $g$ doesn't (though it might still happen that $g(a)=0$), or - and that applies to the OP example - $f$ does not change its sign and hence $g$ changes its sign - which means that $g(a)=0$ and so $a$ is a multiple root of $f$.

Answer 2

If at a point in xy plane first derivative is zero( that is there exists a local extrema), as well as function is zero at that point, we conclude that that point is point of DOUBLE root, that is that root is doubly repeated.

If at a point in xy plane second and first derivative is zero( that is there exists a local extrema, as well as that point is point of infexion), as well as function is zero at that point, we conclude that that point is point of TRIPLE root, that is that root is triply repeated.

In your case since only first dervative is zero at 2 , thus it is point of double root.

Answer 3

Given it's a cubic curve i.e. of degree $3$, its equation must be of the form: $$y=k(x-a)(x-b)(x-c)$$ where $k$ is the leading coefficient and $a,b,c$ are the three roots of the equation and they all are real since two of them are given to be real.

We have to determine the values of the $4$ unknown constants $k,a,b,c,d$ to get the required equation of the curve.

Now you have been given the data about the curve for $3$ points and using the data, you can form $3$ equations. But your original hypothesised equation contains $4$ unknown constants $k,a,b,c,d$.

One can never solve $3$ equations to determine uniquely $4$ unknown constants.

Hence you will never be able to get the equation of the curve using this much data. Or in other words, the question suffers from LACK OF INFORMATION.