Questions: Factoring Out the GCF

Question 1

A student factors 12x⁴ + 8x² and writes 4(3x⁴ + 2x²). Their teacher says this is not fully factored. Why?

• A. The student should have factored out 2 instead of 4, since 2 divides both coefficients
• B. The student only factored out the numerical GCF; the GCF is actually 4x² (including the variable part), giving 4x²(3x² + 2)
• C. The student's answer is correct; 4 is the greatest common factor of 12 and 8
• D. The student should have factored out x⁴ instead, since that is the highest power present

Answer: (B)

Finding the GCF has two parts: the numerical coefficient AND the variable part. The numerical GCF of 12 and 8 is 4 — the student got that right. But the variable GCF uses the lowest power present: both terms have at least x², so x² is also part of the GCF. The complete GCF is 4x², not just 4. Writing 4(3x⁴ + 2x²) is technically factored but not *completely* factored — the leftover expression still has a common factor of x². This is the most common GCF error: remembering the numbers but forgetting the variables.

Question 2

What is the completely factored form of 5x³ + 5x²?

• A. 5(x³ + x²)
• B. 5x(x² + x)
• C. 5x²(x + 1)
• D. x²(5x + 5)

Answer: (C)

The GCF of 5x³ and 5x² is 5x² — the numerical GCF is 5, and the variable GCF is x² (the lowest power). Dividing each term: 5x³ ÷ 5x² = x, and 5x² ÷ 5x² = 1. So the factored form is 5x²(x + 1). Options A and B factor out partial amounts (just 5, or just 5x), leaving the expression incompletely factored. Option D factors out only x², omitting the numerical GCF. The critical check: when one term's entire value equals the GCF, what remains inside the parentheses for that term is 1, not 0.

Question 3

Factoring 6x + 6 as 6(x + 0) is a valid application of GCF factoring because 6 divides both terms.

• T. True
• F. False

Answer: False

When the GCF equals one of the terms exactly, that term leaves a 1 inside the parentheses — not a 0. Here, 6x ÷ 6 = x, and 6 ÷ 6 = 1. The correct factored form is 6(x + 1). Writing 6(x + 0) would expand back to 6x + 0 = 6x, which is not the original expression. This error typically comes from confusing 'the term is gone after factoring' with 'the term becomes 0' — but factoring is the reverse of distribution, and distributing 6 back into 6(x + 1) gives you 6x + 6, confirming the correct answer.

Question 4

You can always verify a GCF factoring by redistributing (expanding) the factored form and confirming it equals the original expression.

• T. True
• F. False

Answer: True

Because GCF factoring is the reverse of the distributive property, expanding the result should always recover the original expression exactly. This check is reliable and fast: 3x²(2x + 3) → 6x³ + 9x² confirms the factoring is correct. If the expansion doesn't match, something went wrong — either the GCF was wrong, or a term inside the parentheses was computed incorrectly. Making this verification a habit catches almost every factoring error.

Question 5

A classmate factors 8x³ + 12x² as 2(4x³ + 6x²). Explain why this is not fully factored, and show the correct completely factored form.

Model answer: The classmate factored out only 2, but the GCF of 8x³ and 12x² is 4x² — both the full numerical GCF (4, not 2) and the variable GCF (x²). To find the correct GCF: the GCF of 8 and 12 is 4; both terms contain at least x², so x² is the variable GCF. Dividing: 8x³ ÷ 4x² = 2x, and 12x² ÷ 4x² = 3. The correct factored form is 4x²(2x + 3). Verify: 4x²(2x + 3) = 8x³ + 12x² ✓

The factoring 2(4x³ + 6x²) is not wrong — it does check out — but it is incomplete. 'Greatest' in GCF means you must factor out the *largest* possible common factor. The expression inside the parentheses, 4x³ + 6x², still has a common factor (2x²), so more factoring is possible. Fully factored means no further common factor remains. The word 'greatest' is doing real work here: any common factor won't do — it must be the greatest one.