Lesson 5: Zeros of Polynomial Functions

Property

Polynomial Equation

An equation of the form $P(x) = 0$ where $P(x)$ is a polynomial function is called a polynomial equation.

Property

The multiplicity of a zero is the number of times the corresponding factor appears in the polynomial's factorization. If $c$ is a zero of multiplicity $m$ in polynomial $P(x) = (x - c)^m \cdot Q(x)$, then:

• Odd multiplicity: graph crosses the x-axis at $x = c$
• Even multiplicity: graph touches but does not cross the x-axis at $x = c$

Examples

Property

When one zero $c$ of a polynomial $P(x)$ is known, synthetic division can be used to divide $P(x)$ by $(x - c)$ to obtain $P(x) = (x - c) \cdot Q(x)$, where $Q(x)$ is a polynomial of lower degree whose zeros are the remaining zeros of $P(x)$.

Examples

Property

The discriminant of a quadratic equation is

1. If $D > 0$, there are two unequal real solutions.
2. If $D = 0$, there is one solution of multiplicity two.
3. If $D < 0$, there are two complex conjugate solutions.

Examples

• For $y = x^2 - x - 3$, the discriminant is $D = (-1)^2 - 4(1)(-3) = 13$. Since $D > 0$, the equation has two distinct real solutions and the graph has two x-intercepts.

• For $y = 2x^2 + x + 1$, the discriminant is $D = 1^2 - 4(2)(1) = -7$. Since $D < 0$, the equation has two complex solutions and the graph has no x-intercepts.