Roots, Zeros and X-Intercepts - Algebra-2

1. Fundamental Concepts

Definition: Roots, zeros, and x-intercepts are points where a polynomial function intersects the x-axis, i.e., where the value of the function is zero.
Linear Factors: Linear factors are expressions of the form (x - c) where c is a root of the polynomial.
Multiplicity: The multiplicity of a root indicates how many times a factor appears in the factored form of the polynomial.

2. Key Concepts

Roots and Zeros: $$(\text{{f}}(x) = 0 \implies x = c$$

X-Intercepts: The x-intercepts are the values of x for which \text{{f}}(x) = 0.

Multiplicity: If a root $c$ has a multiplicity $k$, then $(x - c)^k$ is a factor of the polynomial.

3. Examples

Example 1 (Basic)

Problem: Find the roots of the polynomial $\text{{f}}(x) = x^2 - 5x + 6$.

Step-by-Step Solution:

Factor the polynomial: $\text{{f}}(x) = (x - 2)(x - 3)$.
Set each factor equal to zero: $x - 2 = 0$ and $x - 3 = 0$.
Solve for $x$: $x = 2$ and $x = 3$.

Validation: Substitute $x = 2$ and $x = 3$ into the original equation: $\text{{f}}(2) = 2^2 - 5 \cdot 2 + 6 = 0$, $\text{{f}}(3) = 3^2 - 5 \cdot 3 + 6 = 0$. ✓

Example 2 (Intermediate)

Problem: Determine the roots and their multiplicities for $\text{{f}}(x) = (x - 1)^2(x + 2)$.

Step-by-Step Solution:

The polynomial is already factored: $\text{{f}}(x) = (x - 1)^2(x + 2)$.
Identify the roots and their multiplicities: $x = 1$ with multiplicity 2, and $x = -2$ with multiplicity 1.

Validation: Substitute $x = 1$ and $x = -2$ into the original equation: $\text{{f}}(1) = (1 - 1)^2(1 + 2) = 0$, $\text{{f}}(-2) = (-2 - 1)^2(-2 + 2) = 0$. ✓

4. Problem-Solving Techniques

Factoring Strategy: Always start by factoring the polynomial completely.
Multiplicity Insight: Understand how the multiplicity affects the graph's behavior at the x-axis.
Graphical Interpretation: Use graphs to visualize the relationship between roots and the polynomial’s curve.