Problem 90 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$12 x^{2} y-27 y-4 x^{2}+9$$

Step-by-Step Solution

Answer

The completely factored polynomial is \((3y - 1)(4x^2 - 9)\).

Step 1: Rearrange the Polynomial

Rearrange the terms of the polynomial so that similar terms are together. The polynomial \( 12x^2y - 27y - 4x^2 + 9 \) becomes \(12x^2y - 4x^2 - 27y + 9 \).

Step 2: Factor by Grouping

Now, group the terms and factor out the greatest common factor from each group. We have, \(4x^2(3y - 1) - 9(3y - 1)\).

Step 3: Common Factor

Here, we can see that \(3y - 1\) is a common factor. Hence we can factorize it further to give the final factored form of the polynomial. The polynomial \( 4x^2(3y - 1) - 9(3y - 1)\) becomes \((3y - 1)(4x^2 - 9)\).