Question

1. george factored the polynomial below. his work is shown. did george make a mistake? if so, explain how he could correct his work with at least one complete sentence. problem: $18x^{3}y + 24x^{2}y^{2}+ 6x^{2}y$ answer: $6x^{2}y(3x + 4y)$

Explanation:

Step1: Identify the GCF of all terms

The polynomial is $18x^3y + 24x^2y^2 + 6x^2y$. The greatest common factor (GCF) of the coefficients $18, 24, 6$ is $6$; the GCF of the $x$-terms $x^3, x^2, x^2$ is $x^2$; the GCF of the $y$-terms $y, y^2, y$ is $y$ (note: the third term $6x^2y$ has a $y$ term, so the GCF includes $y$).

Step2: Factor out the GCF from all terms

Factor $6x^2y$ from each term:
$18x^3y = 6x^2y \cdot 3x$
$24x^2y^2 = 6x^2y \cdot 4y$
$6x^2y = 6x^2y \cdot 1$
Combine these: $6x^2y(3x + 4y + 1)$

Step3: Compare to George's answer

George's answer $6x^2y(3x + 4y)$ omits the $+1$ term that comes from factoring the third term $6x^2y$.

Answer:

Yes, George made a mistake. He failed to include the constant term $1$ that results from factoring out the GCF $6x^2y$ from the third term $6x^2y$ of the original polynomial. The correct factored form is $6x^2y(3x + 4y + 1)$.