Math Lesson 11.3.3 - How to list all Possible Zeroes (Roots) of a Polynomial Equation

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Welcome to our Math lesson on How to list all Possible Zeroes (Roots) of a Polynomial Equation, this is the third lesson of our suite of math lessons covering the topic of Solutions for Polynomial Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Listing all Possible Zeroes (Roots) of a Polynomial Equation

As stated in Lesson 11.3.2, the Rational Root Theorem helps identify the possible roots of a polynomial equation. This means it helps narrow the range of possible rational roots but it doesn't give the exact roots of a polynomial equation. If exist, they are found by applying an extra procedure explained below.

Step 1: Identify the constant of the polynomial and write down all its integer factors, including the negative ones. This helps identify the possible values of p.

For example, in the polynomial

the constant of the polynomial is a₀ = -24. The possible integer factors p of this constant are ± 1, ± 2, ± 3, ± 4, ± 8, ± 12 and ± 24.

Step 2: Identify the leading coefficient a_n of the polynomial and write down its integer factors q.

In our polynomial, we have n = 4 and a_n = 2. Hence, the possible values of q are ± 1 and ± 2.

Step 3: Find all possible values of p/q.

In our example, we have

Doing the operations yields

Step 4: Remove all duplicates resulting from step 3.

Thus, removing the duplicates in our example and writing everything in the ascending order yields

This means the list of all possible zeroes of the given polynomial includes the numbers below:

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1. Zeroes Roots Of A Polynomial Equation Feedback. Helps other - Leave a rating for this zeroes roots of a polynomial equation (see below)
2. Polynomials Math tutorial: Solutions for Polynomial Equations. Read the Solutions for Polynomial Equations math tutorial and build your math knowledge of Polynomials
3. Polynomials Video tutorial: Solutions for Polynomial Equations. Watch or listen to the Solutions for Polynomial Equations video tutorial, a useful way to help you revise when travelling to and from school/college
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7. Continuing learning polynomials - read our next math tutorial: Rational Expressions

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