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Prime Polynomials

Part of the Factoring & Quadratics section of Coddy's Algebra journey. Lesson 25 of 63.

Not everything factors. x^2 + x + 1: pairs of 1 are just 1 and 1, summing to 2, not 1. The list is complete, so no whole-number factoring exists. Such a polynomial is called prime, like a prime number: no smaller pieces.

x^2 + 100 is prime too. It LOOKS like the special shape, but the sign is wrong: a SUM of squares, and the sum-times-difference split needs the minus. The pairs of +100 with sum 0 do not exist.

Prime is an answer, not a failure. When the hunt exhausts its complete list, write the expression unchanged and move on with confidence. Knowing that nothing was missed is exactly what the systematic list is for.

Challenge

Easy

One of the shapes from this chapter, hiding in plain sight. Decide whether it factors, then factor it: x^2 - 100.

Try it yourself

x^2 - 100

Factor the expression

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Factoring & Quadratics

1What Factoring Is

Two Directions The Reverse Gear Un-Distributing Letters The Check Is Multiplication Recap: First Factors

4Minus Signs in the Machine

A Negative Product Which One Gets the Minus Both Negative The Sign Map Recap: Signs

7The Zero Product Rule

A Product That Makes Zero One Factor Is Enough The Bare X Every Factor Gets a Turn Recap: Zero Product

10The Quadratic Formula

The Formula, Stated The Number Under the Root Split the Plus-Minus It Always Works Choosing a Method Recap: The Formula

2The Greatest Common Factor

Number and Letter Together The Largest One Higher Powers Three Terms Share Too Recap: The GCF

5Special Shapes

A Missing Middle The Shape in Letters The Perfect Square The Minus VersionPrime PolynomialsRecap: Special Shapes

8Solving by Factoring

Factor, Then Split Signs Under Pressure Arrange It First The GCF Route Bigger Fronts Recap: Solve by Factoring

11How Many Solutions?

Two, One, or None The Double Root None at All Recap: Counting Solutions

3The Sum-Product Machine

Reading the Product Hunting the Pair The Machine in Letters Longer Hunts Recap: The Machine

6Factor Completely

Two Layers Squares Inside Completely Means Completely The Deep Check Recap: Factor Completely

9The Square Root Method

Undoing a Square Arrange, Then Root A Squared Package Roots That Are Not Whole Recap: Square Roots

12Final Challenges

Final: The Full Sweep Final: Factor and Split Final: Arrange First Final: The Rug, Modeled Final: The Rug, Solved Final: No Training Wheels