Maximal Number of Intersections

Algebra Level 3

Suppse $F(x)$ , and $G(x)$ are two distinct (not identical) polynomials, of degrees $f$ and $g$ . What is the maximal possible number of points of intersection between $F(x)$ and $G(x)$ ?

Note: $\text{max}(f,g)$ denotes the higher of the two degrees, while $\text{min}(f,g)$ denotes the lower.

\text{max}(f,g)

\text{max}(f,g)-1

\text{min}(f,g)

\text{min}(f,g)+1

\infty

1 solution

Zane Gates
Jun 8, 2019

Consider the difference between the two polynomials, $D(x)=F(x)-G(x)$ . Then, the degree of $D(x)$ will be at most $\text{max}(f,g)$ . Each root of $D(x)$ corresponds to a point of intersection, so the maximum number of roots is the same as the polynomial's degree, hence $\text{max}(f,g)$ .

Maximal Number of Intersections

1 solution

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