Question

part e: error analysis
a student factored the polynomial below:
$x^3 - 6x^2 - 4x + 24$
student work:
$x^2(x - 6) - 4(x - 6)$
11.

• is the student correct?
• if yes, write the final factored form.
• if no, explain the mistake.

Explanation:

Step1: Verify student's initial factoring

Check grouping step:
Group terms: $(x^3 - 6x^2) + (-4x + 24)$
Factor each group: $x^2(x - 6) - 4(x - 6)$
This matches the student's work, so this step is correct.

Step2: Factor out common binomial

Factor $(x-6)$ from both terms:
$(x - 6)(x^2 - 4)$

Step3: Factor difference of squares

$x^2 - 4$ is a difference of squares, so:
$x^2 - 4 = (x - 2)(x + 2)$

Step4: Combine for full factorization

Substitute back to get the complete factored form:
$(x - 6)(x - 2)(x + 2)$

Answer:

The student's work is partially correct (their initial grouping step is accurate), but they did not fully factor the polynomial. The final completely factored form is $\boldsymbol{(x - 6)(x - 2)(x + 2)}$.