Solve Polynomial Equations by Graphing

Algebra-2

1. Fundamental Concepts

Definition: A polynomial equation is an equation that can be written in the form $P(x) = 0$ , where $P(x)$ is a polynomial.
Graphical Solution: The solutions to a polynomial equation are the x-intercepts of the graph of the corresponding polynomial function.
Multiplicity: The multiplicity of a root affects the behavior of the graph at the x-intercept (e.g., touches and turns, crosses through).

2. Key Concepts

Roots and X-Intercepts: $P(x) = 0 \Rightarrow x$ -intercepts of $y = P(x)$

Multiplicity Behavior: Odd multiplicity: crosses x-axis; Even multiplicity: touches x-axis

Application: Used in physics, engineering, and economics to model real-world phenomena

3. Examples

Example 1 (Basic)

Problem: Solve the polynomial equation by graphing: $x^2 - 4 = 0$

Step-by-Step Solution:

Graph the function $y = x^2 - 4$ .
Identify the x-intercepts where the graph crosses the x-axis.
The x-intercepts are $x = -2$ and $x = 2$ .

Validation: Substitute $x = -2$ and $x = 2$ into the original equation:

$(-2)^2 - 4 = 0 \Rightarrow 4 - 4 = 0 \Rightarrow 0 = 0$ ✓
$2^2 - 4 = 0 \Rightarrow 4 - 4 = 0 \Rightarrow 0 = 0$ ✓

Example 2 (Intermediate)

Problem: Solve the polynomial equation by graphing: $x^3 - 3x^2 - 4x + 12 = 0$

Step-by-Step Solution:

Graph the function $y = x^3 - 3x^2 - 4x + 12$ .
Identify the x-intercepts where the graph crosses or touches the x-axis.
The x-intercepts are $x = -2$ , $x = 2$ , and $x = 3$ .

Validation: Substitute $x = -2$ , $x = 2$ , and $x = 3$ into the original equation:

$(-2)^3 - 3(-2)^2 - 4(-2) + 12 = 0 \Rightarrow -8 - 12 + 8 + 12 = 0 \Rightarrow 0 = 0$ ✓
$2^3 - 3(2)^2 - 4(2) + 12 = 0 \Rightarrow 8 - 12 - 8 + 12 = 0 \Rightarrow 0 = 0$ ✓
$3^3 - 3(3)^2 - 4(3) + 12 = 0 \Rightarrow 27 - 27 - 12 + 12 = 0 \Rightarrow 0 = 0$ ✓

4. Problem-Solving Techniques

Graphing Calculator Use: Utilize graphing calculators to plot the polynomial and find x-intercepts.
Estimation and Zooming: Use the zoom feature to estimate the x-intercepts more accurately.
Table of Values: Create a table of values to identify approximate x-intercepts and confirm with the graph.