Learn on PengienVision, Algebra 2Chapter 3: Polynomial Functions

Lesson 5: Zeros of Polynomial Functions

In this Grade 11 enVision Algebra 2 lesson, students learn how to find and use the zeros of polynomial functions to sketch graphs, applying the Zero-Product Property and the Factor Theorem. The lesson covers multiplicity of a zero and how odd or even multiplicity determines whether a graph crosses or turns at the x-axis. Students also practice finding real and complex zeros using synthetic division and the Quadratic Formula.

Section 1

Solve Polynomial Equations by Factoring

Property

Polynomial Equation

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

Section 2

Multiplicity of Zeros and Graph Behavior

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

Section 3

Synthetic Division to Find Additional Zeros

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

Section 4

The Discriminant

Property

The discriminant of a quadratic equation is

D = b^2 - 4ac

If $D > 0$ , there are two unequal real solutions.
If $D = 0$ , there is one solution of multiplicity two.
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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Solve Polynomial Equations by Factoring

Property

Polynomial Equation

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

Section 2

Multiplicity of Zeros and Graph Behavior

Property

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

Section 3

Synthetic Division to Find Additional Zeros

Property

Examples

Section 4

The Discriminant

Property

The discriminant of a quadratic equation is

D = b^2 - 4ac

If $D > 0$ , there are two unequal real solutions.
If $D = 0$ , there is one solution of multiplicity two.
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.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Solve Polynomial Equations by Factoring

Property

Polynomial Equation

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

Section 2

Multiplicity of Zeros and Graph Behavior

Property

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

Section 3

Synthetic Division to Find Additional Zeros

Property

Examples

Section 4

The Discriminant

Property

The discriminant of a quadratic equation is

D = b^2 - 4ac

If $D > 0$ , there are two unequal real solutions.
If $D = 0$ , there is one solution of multiplicity two.
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.