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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 polynomial is not prime and when completely factored is \((2x + 3)(2x - 3)(3y - 1)\).

Step by step solution

01

Rearrange Terms

First, the given expression needs to be rearranged for common factors to be identifiable more easily. The given expression \(12 x^{2} y-27 y-4 x^{2}+9\) can be rewritten as \(12x^{2}y - 4x^{2} + -27y + 9\).
02

Group Terms

Next, group the terms to factor by grouping. This would give us \((12x^{2}y - 4x^{2}) + (-27y + 9)\).
03

Factor Terms

Now, factor out the GCF from each group. This would give \(4x^{2}(3y - 1) - 9(3y - 1)\).
04

Apply Distributive property

Recognizing that \(3y - 1\) is a common factor in both terms, use the reverse of the Distributive Property (factoring the GCF out) to write this expression in a factored form. \((4x^{2} - 9)(3y - 1)\) is the factored form of the expression.
05

Further Factorization

Observe that \((4x^{2} - 9)\) is the difference of squares and it can be further factored as \((2x + 3)(2x - 3)\). The completely factored form of the expression is \((2x + 3)(2x - 3)(3y - 1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Grouping Method
Factorization using the grouping method involves breaking a polynomial into smaller parts that can be easily factored. When you first look at a polynomial, it might appear complex. The grouping method helps simplify it by making the expression manageable.

First, identify how terms can be paired to have something in common. In the original problem, the polynomial is written as:
  • Group 1: \(12x^{2}y - 4x^{2}\)
  • Group 2: \(-27y + 9\)
You create these groups because each one can be factored separately. After grouping, proceed to "Factor Terms," where each group is handled independently. Stick to pairs that make the rest of the process straightforward.
  • The goal of grouping is to combine pairs that share a common factor.
  • It simplifies the expression and prepares it for deeper factorization.
Difference of Squares
The difference of squares is a specific pattern in polynomials. It is characterized by expressions that resemble: \(a^2 - b^2\). This pattern can be reworked into a product of two binomials: \((a + b)(a - b)\).

In the given problem, once the initial factor of \(3y - 1\) is common from both groups, you have:\((4x^2 - 9)(3y - 1)\). The expression \(4x^2 - 9\) follows the difference of squares formula.

Here's why:
  • Identified like terms: \(a = 2x\) and \(b = 3\).
  • Apply the formula \( (a + b)(a - b) = (2x + 3)(2x - 3) \).
Recognizing these patterns can make large expressions much easier to simplify. Keep an eye out for this structure when factoring polynomials, as it often leads to fast results!
Greatest Common Factor (GCF)
The greatest common factor (GCF) is the highest number or expression that divides all terms in a group. The concept is crucial when factoring, as it simplifies polynomials by extracting common elements.

In factoring by grouping for this problem, identify and factor out the GCF in each group:
  • For \(12x^{2}y - 4x^{2}\), the GCF is \(4x^{2}\).
  • For \(-27y + 9\), the GCF is \(-9\).
By factoring out these common factors, you simplify the expression to:
  • \(4x^2(3y - 1) - 9(3y - 1)\)

This leads to recognizing the common binomial \((3y - 1)\) across both groups. The GCF allows for the isolation of the simplest forms of terms, streamlining factorization processes.

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