Factor Polynomials Game

Factor Polynomials Game
Factoring the Greatest Common Factor (GCF) is the process of finding the largest term that divides evenly into every term of a polynomial. Think of it as the Distributive Property in reverse. Instead of multiplying a term into a group, you are “pulling out” the shared components to see what’s left behind.
 
In this GCF Extractor game, the system generates a polynomial, and you must identify the Greatest Common Factor (GCF) that can be divided out of every part. Scroll down for a detailed explanation.
 

 

How to Play the Factor Polynomials Game
In GCF Extractor, you are a digital technician task with stabilizing “Target Polynomials.” Your mission is to find the Greatest Common Factor (GCF)—the largest value that can be divided out of every term in the expression—and extract it.
Step 1: Isolate the GCF
Look at the Target Polynomial (e.g., 10x⁵ + 15x³). Your first job is to find the Greatest Common Factor.
Find the Coefficient:
What is the largest number that divides evenly into all the numbers in the polynomial?
Example: For 10 and 15, the number is 5.
Find the Exponent:
Look at the x variables. What is the lowest power present in the expression?
Example: Between x⁵ and x³, the lowest is 3.
Enter your findings:
Type 5 in the Coeff box and 3 in the Exp box, then click Extract GCF.

Step 2: Factor the Core
If your GCF is correct, the game will shift to “Step 2: Factor Core.” Now you must determine what remains inside the parentheses after the GCF is removed. Perform Division: Divide every term in the original polynomial by your GCF.
Coefficients:
Divide the numbers (10 ÷ 5 = 2 and 15 ÷ 5 = 3).
Variables:
Subtract the exponents (x^5-3 = x² and x^3-3 = x⁰ or 1).
Select the Result:
You will be presented with three options. Choose the one that matches your division.
Example: The correct choice would be (2x² + 3).

The 3-Step Extraction Process
To factor a polynomial like 12x⁴ + 18x², follow these steps:

1. Find the GCF of the Coefficients (Numbers)
   Look at the numbers (12 and 18). Find the largest number that goes into both.
   Factors of 12: 1, 2, 3, 6, 12
   Factors of 18: 1, 2, 3, 6, 9, 18
   Numerical GCF = 6
   

2. Find the GCF of the Variables (Letters)
   Look at the variables (x⁴ and x²). If every term contains the variable, the GCF is the one with the lowest exponent.
   Comparing x⁴ and x², the lowest power is 2.
   Variable GCF = x²
   

3. Divide and Rewrite
   Combine your results to get the total GCF: 6x². Now, divide every term in the original polynomial by this GCF to find what goes inside the parentheses.
   12x⁴ ÷ 6x² = 2x²
   18x² ÷ 6x² = 3
   Final Factored Form: 6x²(2x² + 3)
   

Tips:
The “Invisible” One: If you see a term like 5x with no visible power, the exponent is actually 1.
Prime Numbers: If the coefficients don’t share any factors other than 1, your GCF coefficient is just 1.
Division Check: Remember that x³ ÷ x³ isn’t nothing; it is 1. If the whole term is extracted, you must keep a “1” in that spot in the core.
Subtraction Rule: When factoring variables, you are always subtracting the GCF’s exponent from the original terms’ exponents.

This video gives a clear, step-by-step approach to simplifying exponential expressions.

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Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

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