FACTORING BY THE GREATEST COMMON FACTOR

Imagine you’re splitting a pizza with friends. You have 12 slices and 18 slices from two different pizzas, and you want to divide them into identical boxes (no cutting slices) so each box has the same number of slices from both pizzas. The greatest common factor (GCF) is the largest number of slices you can put in each box—in this case, 6 (since 6 is the biggest number that divides both 12 and 18 evenly).

Fun Fact:
The GCF is like the "ultimate shared ingredient" in math, whether you’re simplifying recipes, arranging tiles, or even organizing teams!

Question:
If you have 24 chocolates and 36 candies to share equally among friends, what’s the biggest number of friends you can have so everyone gets the same amount of each treat? (Hint: Find the GCF!) 😊

By the end of this explanation, you'll learn:

1. What the Greatest Common Factor (GCF) is—the largest number that divides two or more numbers evenly.

2. How to find the GCF—using simple methods like listing factors or prime factorization.

3. Why it matters—with real-life uses like splitting items equally, simplifying fractions, or solving everyday problems.

Learning Requirements.

Before diving into the Greatest Common Factor (GCF), it’s helpful to already understand these basic math concepts:

1. Factors (Divisors)

• Knowing that a factor of a number divides it exactly (no remainder).
  Example: Factors of 12 are 1, 2, 3, 4, 6, 12.

2. Multiplication & Division

• Comfort with basic multiplication tables and division.
  Example: Recognizing that 18 ÷ 3 = 6 means 3 is a factor of 18.

3. Prime Numbers

• Familiarity with prime numbers (numbers with only two factors: 1 and itself).
  Example: 2, 3, 5, 7, 11, etc.

4. Prime Factorization (Optional but Helpful)

• Breaking down a number into its prime factors (like 12 = 2 × 2 × 3).

Don’t worry if some of these are rusty—we’ll touch on them as needed! 😊

A Quick Prerequisite Recap (30-Second Refresh!)

1. Factors: Numbers that divide evenly into another.

   • Example: Factors of 8 → 1, 2, 4, 8.

2. Prime Numbers: Numbers with only two factors: 1 and itself.

   • Example: 2, 3, 5, 7, 11.

3. Multiplication Basics: Know your times tables (e.g., 3 × 6 = 18).

That’s it! Now you’re ready for GCF. 🚀

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Let me tell you why learning about the Greatest Common Factor (GCF) really matters in math class. It's not just some random skill your teacher makes you learn - this stuff actually helps you become a better problem solver. When you break down those big, scary polynomials by finding the GCF, you're learning to spot patterns and simplify complicated problems. Think of it like taking apart a messy room - first, you group similar items together before you can organize everything properly.

This GCF skill becomes your secret weapon for tougher math down the road. You'll use it when working with quadratic equations, fractions with variables, and algebra problems. But here's the cool part - it's not just about math class. That same logical thinking helps when comparing phone plans to find the best deal, scaling up a recipe when cooking, or figuring out the most efficient way to schedule your day. The GCF process teaches you to look for what different things have in common, which is a smart way to approach all kinds of real-life situations.

What I like about learning GCF is that it gives you a clear, step-by-step method to tackle problems without feeling overwhelmed. Once you get good at it, you start seeing opportunities to simplify things everywhere - in math problems and beyond. It's one of those fundamental skills that makes all the other math you'll learn easier to understand. Teachers don't just teach this to torture you - they know it's a thinking tool you'll use your whole life, even if you don't become a mathematician.

Real-World Applications of Factoring by GCF

Finance & Budgeting

• Interest calculations: Simplifies loan/investment terms
  Example: (6%)(5000) + (6%)(3000) = (6%)(5000 + 3000)

• Cost analysis: Factors common to expenses in business budgets

Engineering & Physics

• Circuit design: Factors affecting resistance in parallel circuits
  Example: (3Ω)(I₁) + (3Ω)(I₂) = (3Ω)(I₁ + I₂)

• Structural equations: Simplifies stress/strain calculations

Computer Science

• Code optimization: Removes redundant operations

• Data compression: Factors repeating patterns in files

Chemistry

• Balance equations: Factors common coefficients
  Example: (2)(H₂) + (2)(O₂) = (2)(H₂ + O₂)

Everyday Life

• Time management: Group similar tasks

• Cooking: Scales recipes using common factors
  Example: (3)(2cups) + (3)(1cup) = (3)(2 + 1)cups

Practical Example:

A landscaper needs: (15)(plants) for Project A + (25)(plants) for Project B

GCF solution: (5)(3 + 5)plants → Orders in 5-plant bundles

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Activity: "GCF Mission Possible"

Group yourselves into: 3–4 students
Time: 10–15 minutes
Materials: Paper, markers/pens, and (optional) small manipulatives (e.g., beads, tiles).

Activity Title: The Snack Attack Challenge

Problem:
Your group has 24 cookies and 36 candies. What’s the largest number of friends you can share them with equally (no leftovers)?

Tasks:

1. List the factors of 24 and 36 separately.

2. Circle the common factors—then pick the greatest one.

3. Verify: Divide both numbers by the GCF. Do they split evenly?

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Factoring by the Greatest Common Factor (GCF) is the process of breaking down an algebraic expression into a product of simpler terms by extracting the largest shared factor.

Key Steps:

1. Step 1: Identify the GCF of the coefficients and variables.

• Numbers: Find the largest number dividing all coefficients (e.g., GCF of 12 and 18 is 6).

• Variables: Take the variable with the smallest exponent (e.g., GCF of $x^3$ and $x^2$ is $x^2$).

• Step 2: Rewrite each term as GCF × remaining factor.

• Step 3: Factor out the GCF using the distributive property:

  $$\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>a</b></span></span>}\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>b</b></span></span>}<span\; style="background-color:\; \#fcff01;"><span><b>+</b></span></span>\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>a</b></span></span>}\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>c</b></span></span>}<span\; style="background-color:\; \#fcff01;"><span><b>=</b></span></span>\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>a</b></span></span>}<span\; style="background-color:\; \#fcff01;"><span><b>(</b></span></span>\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>b</b></span></span>}<span\; style="background-color:\; \#fcff01;"><span><b>+</b></span></span>\mathrm{<span\; style="background-color:\; \#fcff01;"><span><b>c</b></span></span>}<span\; style="background-color:\; \#fcff01;"><span><b>)</b></span></span>$$

Example:

Factor $\mathrm{<span\; style="background-color:\; \#04ff00;">8</span>}{\mathrm{<span\; style="background-color:\; \#04ff00;">x</span>}}^{\mathrm{<span\; style="background-color:\; \#04ff00;">3</span>}}<span\; style="background-color:\; \#04ff00;">+</span>\mathrm{<span\; style="background-color:\; \#04ff00;">12</span>}{\mathrm{<span\; style="background-color:\; \#04ff00;">x</span>}}^{\mathrm{<span\; style="background-color:\; \#04ff00;">2</span>}}$:

• GCF of coefficients (8, 12): 4

• GCF of variables ($x^3,x^2$): ${\mathrm{<span\; style="background-color:\; \#fcff01;">x</span>}}^{\mathrm{<span\; style="background-color:\; \#fcff01;">2</span>}}$
  

• Total GCF: $\mathrm{<span\; style="background-color:\; \#fcff01;">4</span>}{\mathrm{<span\; style="background-color:\; \#fcff01;">x</span>}}^{\mathrm{<span\; style="background-color:\; \#fcff01;">2</span>}}$
  

• Factored form: 

  $$\mathrm{<b><span\; style="background-color:\; \#04ff00;">8</span></b>}{\mathrm{<b><span\; style="background-color:\; \#04ff00;">x</span></b>}}^{\mathrm{<b><span\; style="background-color:\; \#04ff00;">3</span></b>}}<b><span\; style="background-color:\; \#04ff00;">+</span></b>\mathrm{<b><span\; style="background-color:\; \#04ff00;">12</span></b>}{\mathrm{<b><span\; style="background-color:\; \#04ff00;">x</span></b>}}^{\mathrm{<b><span\; style="background-color:\; \#04ff00;">2</span></b>}}<b><span\; style="background-color:\; \#04ff00;">=</span></b>\mathrm{<b><span\; style="background-color:\; \#04ff00;">4</span></b>}{\mathrm{<b><span\; style="background-color:\; \#04ff00;">x</span></b>}}^{\mathrm{<b><span\; style="background-color:\; \#04ff00;">2</span></b>}}<b><span\; style="background-color:\; \#04ff00;">(</span></b>\mathrm{<b><span\; style="background-color:\; \#04ff00;">2</span></b>}\mathrm{<b><span\; style="background-color:\; \#04ff00;">x</span></b>}<b><span\; style="background-color:\; \#04ff00;">+</span></b>\mathrm{<b><span\; style="background-color:\; \#04ff00;">3</span></b>}<b><span\; style="background-color:\; \#04ff00;">)</span></b>$$
  

Detailed Illustration:

Factoring by Greatest Common Factor (GCF) 

Problem 1: Factor the expression 6x² + 9x completely

Follow the Step-by-Step Solution:

Step 1: Find the GCF of the coefficients

• Numbers: 6 and 9

• Factors of 6: (1)(6), (2)(3)

• Factors of 9: (1)(9), (3)(3)

• → GCF is 3

Step 2: Find the GCF of the variables

• Terms: x² and x

• x² = (x)(x)

• x = (x)(1)

• → GCF is x

Step 3: Determine the total GCF

• Multiply the coefficient and variable GCFs

• Total GCF = (3)(x) = 3x

Step 4: Rewrite each term

• 6x² = (3x)(2x)

• 9x = (3x)(3)

Step 5: Factor out the GCF

• 6x² + 9x = (3x)(2x) + (3x)(3)

• = 3x(2x + 3)

Now, the final factored form of 6x² + 9x  is 3x(2x + 3), or 6x² + 9x = 3x(2x + 3)

To check:

Distribute 3x to (2x + 3)
3x(2x + 3) = (3x)(2x) + (3x)(3) = 6x² + 9x ✔️

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Problem 2: Factor 4y³ + 10y. Follow the process shown in problem 1.

Step 1: GCF of 4 and 10 = 2

Step 2: GCF of y³ and y = y

Step 3: Total GCF = (2)(y)

Step 4: Rewrite: 4y³ = (2y)(2y²). For 10y , the factor is (2y)(5) or 10y = (2y)(5)

Step 5: The factors are: 2y and (2y² + 5)

Now, the final factored form of 4y³ + 10y is 2y(2y² + 5), or 4y³ + 10y = 2y(2y² + 5).

To check:

Distribute 2y to (2y² + 5)
2y(2y² + 5) =(2y)(2y²) + (2y)(5) = 4y³ + 10y ✔️

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Problem 3: Factor 12a²b⁴ - 18ab³

Step 1: GCF of 12 and 18 = 6

Step 2: GCF of variables: a² and a = a, for b⁴ and b³ = b³

Step 3: Total GCF = (6)(a)(b³), 

Step 4: Rewrite: 12a²b⁴ = (6ab³)(2ab) , and -18ab³ = (-3)(6ab³), or 

Step 5: Factor: 6ab³(2ab - 3)

Final Answer: 12a²b⁴ - 18ab³ = (6ab³)(2ab - 3)

To check:

Distribute (6ab³) to (2ab - 3)

 (6ab³)(2ab - 3) =  (6ab³)(2ab) -3 (6ab³) =12a²b⁴ - 18ab³  ✔️

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Problem 4: Negative Leading Term

Factor -5x⁴ + 15x²

Step 1: GCF of 5 and 15 = 5

Step 2: GCF of x⁴ and x² = x²

Step 3: Factor out -5x² (include negative)

Step 4: Rewrite: for (-5x⁴ )= (-5x²)(x²), and for (15x²) = (-5x²)(-3)

Step 5: Factor: -5x²(x² - 3)

Final Answer: -5x⁴ + 15x² = -5x²(x² - 3)

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Problem 5: Factor 8m³n + 12m²n² - 20mn³

Step 1: GCF of 8,12,20 = 4

Step 2: GCF of variables: m³,m²,m = m, n,n²,n³ = n

Step 3: Total GCF = (4)(m)(n)

Step 4: Rewrite: 

        8m³n = (4mn)(2m²), 

     12m²n² = (4mn)(3mn)
     -20mn³ = (4mn)(-5n²)

Step 5: Factor: 4mn(2m² + 3mn - 5n²)

Final Answer: 8m³n + 12m²n² - 20mn³ = 4mn(2m² + 3mn - 5n²)

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Problem 6:  Factor 21 + 14 - 35. This is literally the simplest.

Step 1: GCF of 21,14,35 = 7

Step 2: No variables present

Step 3: Rewrite:
          21 = (7)(3)
          14 = (7)(2)
         -35 = (7)(-5)

Step 4: Factor: 7(3 + 2 - 5)

Final Answer: 21 + 14 - 35 = 7(3 + 2 - 5)

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NOTE! The next examples have missing items for you to complete. Kindly hit the word click and write the missing part.

Problem 7:: Simple GCF with Integers and Variables$\mathrm{<span\; style="background-color:\; \#04ff00;">6x³\; +\; 9x²</span>}$

Expression:

Step 1: Identify the GCF of the coefficients: The GCF of 6 and 9 is click ?.

Step 2: Identify the GCF of variables: The Lowest power of $x$ in both terms is $x^2$.

Step 3:Factor out the GCF ($3x^2$): 6x³ + 9x² = 3x²(2x) + 3x²(3) = 3x²(2x + 3)

Step 4:Final Factored Form: $3x^2(2x+3)$

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Problem 8: GCF with Negative Coefficients

Expression: -4y⁴ - 8y²

Step 1: Find the GCF of the coefficients (-4,-8): -4

Step 2: Find the GCF of variables (y⁴,y²): y²

Step 3:Factor out -4y²: So, -4y⁴ - 8y² = -4y²(y²) + (-4y²)(2) = click (y² + 2)

So the Factored form of 4y⁴ - 8y² is -4y²(y² + 2), or 4y⁴ - 8y² = -4y²(y² + 2)

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Problem 9: GCF with Parentheses (Grouping)

Given expression: 5x(x+3) - 2(x+3)

Step 1:Identify common binomial factor: (x+3)

Step 2: Factor out (x+3): so, 5x(x+3) - 2(x+3) = click(5x - 2)

Now the  factored form of 5x(x+3) - 2(x+3) is (x+3)(5x-2), or 5x(x+3) - 2(x+3) = (x+3)(5x-2)

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Problem 10: GCF with Multiple Variables

Given the expression: 12a²b³ + 18ab⁴ - 24a³b²

Step 1: Find the GCF of the coefficients (12,18,24): 6

Step 2: Find GCF of variables (a²b³,ab⁴,a³b²): ab²

Step 3: Factor out 6ab²:

12a²b³ + 18ab⁴ - 24a³b² = 6ab²(2ab) + 6ab²(3b²) + 6ab²(-4a²) = 6ab²(2ab + 3b² - 4a²)

Now the factored form of 12a²b³ + 18ab⁴ - 24a³b² is 6ab²(2ab + 3b² - 4a²), or

12a²b³ + 18ab⁴ - 24a³b² = 6ab²(2ab + 3b² - 4a²)

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Share it with your classmates and work collaboratively.

EXERCISES: The given expressions were solved by partial solution. Fill it up by clicking the blanks.

Problem1:SimpleGCF

Expression:12x³+18x²

1.GCFof coefficients(12,18):Click here to answer 

2.GCFof variables(x³,x²):Click here to answer

3. Factor out the GCF:

12x³+18x² = (2x) + (3) = (+)

Factored form: Click here to answer

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Problem2:NegativeCoefficients

Expression:-9y⁵-15y³

1.GCFof coefficients(-9,-15):Click here to answer

2.GCFof variables(y⁵,y³):Click here to answer

3. Factor out the GCF:

-9y⁵-15y³ = (3y²) + (5) = (+)

Factored form: Click here to answer

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Problem3:MultipleVariables

Expression:24a⁴b²-36a³b³+60a²b⁴

1.GCFof coefficients(24,36,60):Click here to answer

2.GCFof variables(a⁴b²,a³b³,a²b⁴):Click here to answer

3. Factor out the GCF:

24a⁴b²-36a³b³+60a²b⁴=( ) + ( ) + ( )=( Click here to answer )

Factored form: Click here to answer

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Problem4:Grouping

Expression:5x(y+7)-2(y+7)

1. Common binomial factor: Click here to answer to answer

2.Factorout thebinomial:

5x(y+7)-2(y+7)=(-)Click here to answer

Factored form: Click here to answer

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Problem5:Mixed Practice

Expression:48m⁵n³-64m⁴n⁴+80m³n⁵

1.GCFof coefficients(48,64,80):Click here to answer

2.GCFof variables(m⁵n³,m⁴n⁴,m³n⁵):Click here to answer

3. Factor out the GCF:

48m⁵n³-64m⁴n⁴+80m³n⁵=( ) + ( ) + ( )=(Click here to answer)

Factored form: Click here to answer
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Answer to exercises;

Problem1:SimpleGCF
Expression:12x³+18x²
1.GCFofcoefficients(12,18):6
2.GCFofvariables(x³,x²):x²
3. Factor out the GCF:
12x³+18x²=6x²(2x)+6x²(3)=6x²(2x+3)
Factoredform:6x²(2x+3)

Problem2:NegativeCoefficients
Expression:-9y⁵-15y³
1.GCFofcoefficients(-9,-15):-3
2.GCFofvariables(y⁵,y³):y³
3. Factor out the GCF:
-9y⁵-15y³=-3y³(3y²)+-3y³(5)=-3y³(3y²+5)
Factoredform:-3y³(3y²+5)

Problem3:MultipleVariables
Expression:24a⁴b²-36a³b³+60a²b⁴
1.GCFofcoefficients(24,36,60):12
2.GCFofvariables(a⁴b²,a³b³,a²b⁴):a²b²
3. Factor out the GCF:
24a⁴b²-36a³b³+60a²b⁴=12a²b²(2a²)+12a²b²(-3ab)+12a²b²(5b²)=12a²b²(2a²-3ab+5b²)
Factoredform:12a²b²(2a²-3ab+5b²)

Problem4:Grouping
Expression:5x(y+7)-2(y+7)
1.Commonbinomialfactor:(y+7)
2.Factoroutthebinomial:
5x(y+7)-2(y+7)=(y+7)(5x-2)
Factoredform:(y+7)(5x-2)

Problem5:MixedPractice
Expression:48m⁵n³-64m⁴n⁴+80m³n⁵
1.GCFofcoefficients(48,64,80):16
2.GCFofvariables(m⁵n³,m⁴n⁴,m³n⁵):m³n³
3. Factor out the GCF:
48m⁵n³-64m⁴n⁴+80m³n⁵=16m³n³(3m²)+16m³n³(-4mn)+16m³n³(5n²)=16m³n³(3m²-4mn+5n²)
Factoredform:16m³n³(3m²-4mn+5n²)

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