Factoring Numerical & Algebraic Expressions - Sixth Grade Mathematics
Factoring reverses the distributive property, decomposing sums and differences into products of their Greatest Common Factor and a simplified grouped expression.
What is Factoring?
Factoring is the exact inverse operation of distribution. While distribution expands an expression by multiplying a factor across parentheses, factoring packages an expanded expression back into parentheses by extracting the Greatest Common Factor (GCF).
Distribution (Expanding): 4 * (x + 3) --> 4x + 12
Factoring (Reversing): 4x + 12 --> 4 * (x + 3)
The Four-Step Factoring Protocol
Factoring an algebraic expression follows a four-step sequence.
+-------------------------------------------------------------------------+
| ALGEBRAIC FACTORING PROTOCOL |
| |
| Step 1: Identify all terms in the expression. |
| Step 2: Find the Greatest Common Factor (GCF) of the numerical |
| coefficients and constants. |
| Step 3: Write the GCF OUTSIDE a pair of parentheses. |
| Step 4: Divide EACH original term by the GCF to determine the terms |
| that remain inside the parentheses: (Term / GCF). |
+-------------------------------------------------------------------------+
Worked Example: Factor the expression 18x + 24
Step 1: Identify terms: 18x and 24.
Step 2: Find GCF of 18 and 24:
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
GCF = 6.
Step 3: Write GCF outside: 6( ... )
Step 4: Divide each term by 6:
18x / 6 = 3x
24 / 6 = 4
Result: 6(3x + 4).
Factoring Numerical Expressions (Using GCF)
The distributive property can also factor purely numerical sums, an essential technique for mental arithmetic and fraction simplification.
Factor the numerical sum: 42 + 56
Step 1: Find GCF(42, 56):
42 = 6 * 7 = 2 * 3 * 7
56 = 8 * 7 = 2^3 * 7
GCF = 2 * 7 = 14.
Step 2: Divide each addend by 14:
42 / 14 = 3
56 / 14 = 4
Factored Form: 14 * (3 + 4).
Verification: 14 * 7 = 98; and 42 + 56 = 98. (Confirmed!)
Verifying Factored Expressions
Always check your factored result by expanding it back out using the distributive property. If distributing the GCF reproduces your exact starting expression, your factorization is complete and correct. For 6(3x + 4): 6 * 3x = 18x, and 6 * 4 = 24; 18x + 24 matches the original expression.
Exercises
Exercise 1. Factor each algebraic expression using the Greatest Common Factor: (a) 15x + 35; (b) 24y - 16; (c) 12n + 18; (d) 30k - 45.
Exercise 2. Use the distributive property to factor the sum of two whole numbers: 36 + 60. Write the factored expression using the GCF.
Exercise 3. A student factors 20x + 30 as 5(4x + 6). Explain why this factorization is incomplete, and provide the fully factored expression.
Solutions
Solution to Exercise 1. For (a) 15x + 35: GCF of 15 and 35 is 5; divide: 15x / 5 = 3x; 35 / 5 = 7; factored form is 5(3x + 7). For (b) 24y - 16: GCF of 24 and 16 is 8; divide: 24y / 8 = 3y; 16 / 8 = 2; factored form is 8(3y - 2). For (c) 12n + 18: GCF of 12 and 18 is 6; divide: 12n / 6 = 2n; 18 / 6 = 3; factored form is 6(2n + 3). For (d) 30k - 45: GCF of 30 and 45 is 15; divide: 30k / 15 = 2k; 45 / 15 = 3; factored form is 15(2k - 3).
Solution to Exercise 2. Find GCF of 36 and 60: prime factorization of 36 is 2^2 * 3^2; prime factorization of 60 is 2^2 * 3 * 5; GCF = 2^2 * 3 = 12. Divide each addend by 12: 36 / 12 = 3, and 60 / 12 = 5. Factored expression: 12(3 + 5).
Solution to Exercise 3. The factorization 5(4x + 6) is incomplete because the terms inside the parentheses (4x and 6) still share a common factor of 2. A factorization is complete only when the Greatest Common Factor has been extracted and the remaining terms are co-prime. The true GCF of 20 and 30 is 10. Dividing both terms by 10 gives 20x / 10 = 2x and 30 / 10 = 3. The fully factored expression is 10(2x + 3).