1. ## [SOLVED] Polynomial Problem

Consider two distinct polynomials of degree N:

A(X) = a0 + a1*X + a2*X^2 + … + aN*X^N

B(X) = b0 + b1*X + b2*X^2 + … + bN*X^N

where ak and bk are real numbers. Suppose that at X = 1, 2, ..., N, the graphs of A(X) and B(X) intersect.

Questions:

1.) At what places besides X = 1, 2, ..., N do the graphs of A and B intersect?

2.) What is (a0 – b0)/(aN – bN) ?

2. Originally Posted by dxdy
Consider two distinct polynomials of degree N:

A(X) = a0 + a1*X + a2*X^2 + … + aN*X^N

B(X) = b0 + b1*X + b2*X^2 + … + bN*X^N

where ak and bk are real numbers. Suppose that at X = 1, 2, ..., N, the graphs of A(X) and B(X) intersect.

Questions:

1.) At what places besides X = 1, 2, ..., N do the graphs of A and B intersect?
they can't intersect in anywhere else because otherwise $A(x)-B(x)$ must be a polynomial of degree at least $n+1$ and that is obviously false.

2.) What is (a0 – b0)/(aN – bN) ?
$A(x)-B(x)=(a_n - b_n)x^n + \cdots + a_0 - b_0$ and $x=1,2, \cdots , n$ are the roots of $A(x)-B(x).$ therefore: $\frac{a_0 - b_0}{a_n - b_n}=(-1)^n n!.$