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Chapter 0: Problem 90

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

Short Answer

Expert verified
The completely factored form of the polynom \(12x^2y - 27y - 4x^2 + 9\) is \((2x -3)(2x + 3)(3y - 1)\).

Step by step solution

01

Rewrite the Expression

First, rearrange the expression to get similar terms together. Here, the order of the polynom also matter. We get \(12x^2y - 4x^2 - 27y + 9\)
02

Factor by Grouping

Then perform factoring by grouping. Factoring by grouping involves separating the polynom in two groups and factor out the greatest common factor from each group. The two groups here will be \(12x^2y - 4x^2\) and \(-27y + 9\). Now the expression becomes \(4x^2(3y - 1) -9(3y - 1)\)
03

Factor out the Common Binomial

There is a common factor in the two terms: \(3y - 1\). It can be factored out to obtain the fully factored form, which gives \( (4x^2 - 9) (3y - 1)\)
04

Simplify further if possible

The expression \(4x^2 - 9\) is a difference of squares and can be further simplified into \((2x -3)(2x + 3)\). Hence, the complete factorization is \((2x -3)(2x + 3)(3y - 1)\).

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