Zeros of Polynomial Functions
To determine where a function [latex]f[/latex] intersects the [latex]x[/latex]-axis, we need to solve the equation [latex]f(x)=0[/latex] for [latex]x[/latex].
In the case of the linear function [latex]f(x)=mx+b[/latex], the [latex]x[/latex]-intercept is given by solving the equation [latex]mx+b=0[/latex]. Which can be found by [latex](−\frac{b}{m},0)[/latex].
In the case of a quadratic function, finding the [latex]x[/latex]-intercept(s) requires finding the zeros of a quadratic equation: [latex]ax^2+bx+c=0[/latex]. In some cases, it is easy to factor the polynomial [latex]ax^2+bx+c[/latex] to find the zeros. If not, we make use of the quadratic formula.
The Quadratic Formula
Consider the quadratic equation
where [latex]a\ne 0[/latex]. The solutions of this equation are given by the quadratic formula
The discriminant, given by, [latex]b^2-4ac[/latex], determines the nature of a quadratic equation’s solutions. A positive discriminant indicates two distinct real solutions, a discriminant of zero results in exactly one real solution, and a negative discriminant means the equation has no real solutions.
In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the [latex]x[/latex]-axis. For this content, we will only focus on finding the zeros of quadratic polynomials.
Consider the quadratic function [latex]f(x)=3x^2-6x+2[/latex]. Find the zeros of [latex]f(x)[/latex].