Expert: Josh - 7/23/2008

Question
two polynomials can have the same x intercept.Explain.  Why

Answer
Hello Samreen,

I am trying to see what the question is really asking to offer you an explanation. Obviously, if two polynomials are the same, then they must have the same x-intercept (if they cross the x-axis at all).

A number of factors make this question somewhat vague. Firstly, the polynomial space consists of a large family of curves, including those those which we are familiar with, such as linear ones, quadratics, cubics and so forth.

If we start with the simplest case, a linear equation like y=mx+b, where m is the gradient and b is the y-intercept, to find a solution in x, we must set y=0. In doing so, we remove one degree of freedom from the polynomial, making it more constrained. We end up with 0=mx+b, and x=-b/m.

Now, consider two straight lines described by y=mx+b and y=nx+c. To make each equation unique, let's assume that the slope "m" is different from "n". To formally answer the question you posed, we ask: can we select a value for "c" in the second polynomial such that the x-intercept appears at x=-b/m, for some fixed values of b, m and n.

As always, y=nx+c has a solution at x=-c/n, provided n is not zero (the line is not parallel to the x-axis). Equating this to x=-b/m, we find that -c/n=-b/m, when c=n*(b/m).

Recapping what we have done, we started with a linear equation y=nx+c which contains two free parameters "n" and "c". To force this line to pass through x=-b/m (the x-intercept from the first equation), we effectively remove one degree of freedom (see note 1).  But, we are at liberty to pick c=n*(b/m) to ensure that the two equations are different and the condition (viz., same x-intercept) is satisfied.

Note 1: To see this, observe that the expression of c [c=n*(b/m)] now involves n. Once "n" is set, "c" is no longer an arbitrary constant. We cannot set "c" to anything we want; "c" has to be n*b/m.

I think the question should be "why two polynomials can have the same x-intercept without being identical, assuming a solution in x exists". I hope I answered your question.

Regards
Josh