Finding All Zeros of Polynomials

This detailed guide explains the fundamental concepts and steps for identifying rational and irrational zeros of polynomials. The content covers essential methods and techniques for solving polynomial equations.

Definition: Real zeros are any zeros that do not contain an imaginary component $i$, while rational zeros are those that can be expressed as whole numbers or proper fractions.

Vocabulary: Irrational zeros are numbers that are non-repeating, non-terminating, or contain square roots, while imaginary zeros contain the imaginary unit i and do not intersect with the x-axis.

The systematic approach to finding all zeros involves:

Highlight: The key steps include:

1. Entering the polynomial in a graphing calculator and finding rational zeros through intersection
2. Using synthetic division with rational zeros
3. Continuing synthetic division until reaching a linear or quadratic equation
4. Using the quadratic formula when necessary
5. Verifying that the number of zeros equals the polynomial's degree

Example: Two detailed problems demonstrate the process:

1. P$x$ = 4x³-17x-2, which yields both rational and irrational zeros
2. P$x$ = 3x⁴+6x³+9x²+26x+8, showing how to handle higher-degree polynomials

Quote: "Goal-To get quotient down to either a LINEAR EQUATION or QUADRATIC EQUATION."