Question

factor the four - term polynomial by grouping.
12x²y - 18x² - 18y + 27
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. 12x²y - 18x² - 18y + 27 =

b. the polynomial is not factorable by grouping.

Explanation:

Step1: Rearrange the polynomial

First, rearrange the terms of the polynomial \(12x^{2}y - 18x^{2}-18y + 27\) to group terms with common factors. We can group the first two terms and the last two terms: \((12x^{2}y - 18x^{2})+(-18y + 27)\)

Step2: Factor out GCF from each group

For the first group \(12x^{2}y - 18x^{2}\), the greatest common factor (GCF) is \(6x^{2}\). Factoring out \(6x^{2}\), we get \(6x^{2}(2y - 3)\).

For the second group \(-18y + 27\), the GCF is \(-9\) (or \(9\), but factoring out \(-9\) will help in getting a common binomial factor). Factoring out \(-9\), we get \(-9(2y - 3)\).

So now the polynomial becomes \(6x^{2}(2y - 3)-9(2y - 3)\)

Step3: Factor out the common binomial factor

Now, we can see that both terms have a common binomial factor of \((2y - 3)\). Factoring out \((2y - 3)\), we get \((2y - 3)(6x^{2}-9)\)

Step4: Factor out GCF from the second factor

We can further factor \(6x^{2}-9\) by factoring out the GCF of \(3\). So \(6x^{2}-9 = 3(2x^{2}-3)\)

Putting it all together, the factored form is \(3(2y - 3)(2x^{2}-3)\) (or we can also write it as \(3(2y - 3)(2x^{2}-3)\) or factor differently, but the key is that the polynomial is factorable by grouping)

Wait, but let's check the original options. Option A says to fill in the box and Option B says it's not factorable. But from our steps, we can factor it. Wait, maybe I made a mistake in grouping. Let's re - group.

Alternative grouping: Let's group \(12x^{2}y-18y\) and \(-18x^{2}+27\)

First group: \(12x^{2}y - 18y=6y(2x^{2}-3)\)

Second group: \(-18x^{2}+27=-9(2x^{2}-3)\)

Now the polynomial is \(6y(2x^{2}-3)-9(2x^{2}-3)=(2x^{2}-3)(6y - 9)=3(2x^{2}-3)(2y - 3)\), which is the same as before. So the polynomial is factorable by grouping. So Option A is correct and the factored form is \(3(2x^{2}-3)(2y - 3)\) (or other equivalent forms)

Answer:

A. \(12x^{2}y - 18x^{2}-18y + 27=\boldsymbol{3(2x^{2}-3)(2y - 3)}\) (or equivalent factored forms like \((2y - 3)(6x^{2}-9)\) which can be further factored as \(3(2y - 3)(2x^{2}-3)\))