Questions: Factoring Completely

Question 1

A student factors 4x³ − 16x and writes 4x(x² − 4), then stops. Which statement best describes this work?

• A. It is fully factored — the polynomial is written as a product of simpler expressions
• B. It is incorrect — the GCF should have been 2x, not 4x
• C. It is incomplete — (x² − 4) is a difference of squares that can be factored further into (x + 2)(x − 2)
• D. It is incomplete — the GCF was not factored out before the trinomial step

Answer: (C)

This is the classic 'stopping too early' error. 4x(x² − 4) is partially factored — the GCF was correctly identified and removed. But x² − 4 is a difference of squares: a² − b² = (a + b)(a − b), so x² − 4 = (x + 2)(x − 2). The fully factored form is 4x(x + 2)(x − 2). A polynomial is only factored completely when no individual factor can be factored further — you must check every factor at every step.

Question 2

Which of the following expressions is fully factored?

• A. 3x(x² − 9)
• B. (x + 3)(x² − 9)
• C. 3x(x + 3)(x − 3)
• D. 3x³ − 27x

Answer: (C)

Option C, 3x(x + 3)(x − 3), is the only fully factored form. Starting from 3x³ − 27x: factor out the GCF of 3x to get 3x(x² − 9). Then x² − 9 is a difference of squares that factors into (x + 3)(x − 3). Option A still has x² − 9 unfactored. Option B has (x² − 9) unfactored and the GCF wasn't addressed. Option D is the original unfactored expression.

Question 3

When factoring a polynomial largely, you primarily need to check whether each resulting factor can be factored further if the original polynomial has four or more terms.

• T. True
• F. False

Answer: False

You must inspect every factor at every step regardless of the original polynomial's term count. A two-term polynomial like 2x² − 8 first yields 2(x² − 4) after the GCF step, but x² − 4 is still a difference of squares that must be factored to (x + 2)(x − 2). The check 'can this factor be factored further?' applies universally — the number of terms in the original polynomial is irrelevant once you are past the initial step.

Question 4

Factoring out the GCF as the first step in factoring completely typically simplifies the remaining factoring task.

• T. True
• F. False

Answer: True

The GCF step is not just convention — it actively reduces the complexity of what follows. If you have 6x³ − 24x and skip the GCF, you face a cubic with no obvious pattern. Factor out 6x first and you get 6x(x² − 4) — a recognizable difference of squares. Similarly, pulling out a numeric GCF (like 3) can transform a trinomial with a large leading coefficient into one with leading coefficient 1, making the trinomial factoring step straightforward. GCF first is a strategic simplification, not a formality.

Question 5

Explain why the strategy for factoring completely always begins with the GCF step, even when the GCF is just a number like 2 or 3. What would happen if you skipped this step?

Model answer: Factoring out the GCF first reduces the degree and size of coefficients in the remaining expression, making subsequent steps much easier. If the GCF is a monomial like 3x, pulling it out converts a cubic into a quadratic; if it's just a number like 2, it simplifies the coefficients. Skipping the GCF forces you to work with larger, more complex expressions. For example, 2x² − 8 without the GCF step requires recognizing 2x² − 8 directly as a pattern, which is harder. After factoring out 2, you see x² − 4 — immediately a difference of squares. More importantly, skipping the GCF can hide the complete factored form entirely: you might successfully factor x² − 4 into (x + 2)(x − 2) but miss the factor of 2, leaving your answer incomplete.

The GCF step is strategic, not ceremonial. It applies to every polynomial (the GCF might be 1, in which case no simplification occurs, but you've confirmed that). Teaching GCF-first as a discipline prevents the most common factoring errors: working with unnecessarily large coefficients and leaving the GCF hidden inside a factor.