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What is the completely factored form of $12 x y-9 x-8 y+6$ ?


A$(3 x-2)(4 y-3)$
B$3 x(4 y-3)-2(4 y+3)$
C$-6 x(4 y-3)(4 y-3)$
D$(3 x-2)(4 y-3)(4 y-3)$

Verified Answer :

  • Group the four terms into two pairs: \((12xy - 9x)\) and \((-8y + 6)\).
  • Extract the common factor from each pair: \(3x(4y - 3) - 2(4y - 3)\).
  • Factor out the common binomial \((4y - 3)\) to arrive at the fully factored expression \((3x - 2)(4y - 3)\).
  • The final answer is \(\boxed{(3x - 2)(4y - 3)}\).

Explanation

  1. Analysis of the Expression

The given expression is \(12xy - 9x - 8y + 6\). To factor this expression, we observe that it has four terms. A common and effective strategy for such expressions is grouping, where we arrange the terms into two pairs and find common factors within each pair. The goal is to see if a common binomial factor emerges after factoring out the greatest common factor (GCF) from each group.

  1. Grouping and Factoring Each Part

We can group the four terms as follows: \((12xy - 9x)\) and \((-8y + 6)\). Now, we factor out the GCF from each group:

  • From the first group \((12xy - 9x)\), the greatest common factor is \(3x\). Factoring it out leaves us with \(3x(4y - 3)\).
  • From the second group \((-8y + 6)\), the greatest common factor is \(-2\). Factoring it out leaves us with \(-2(4y - 3)\).

Combining these results, the original expression can be rewritten as:
\[3x(4y - 3) - 2(4y - 3)\]

  1. Final Factorization

Now, we observe that the binomial \((4y - 3)\) is a common factor in both terms of our expression. We can factor out \((4y - 3)\) from the entire expression:
\[(3x - 2)(4y - 3)\]
Thus, the completely factored form of \(12xy - 9x - 8y + 6\) is \((3x - 2)(4y - 3)\). This corresponds to the first option provided.

Examples

Factoring by grouping is like sorting items into bins to see what patterns emerge. If you are organizing a shelf with items labeled by type and size—like 12 large blue blocks and 9 small blue blocks, minus 8 large red blocks and 6 small red blocks—grouping helps you see the common factors, such as the color or size, making it much easier to count or rearrange them. This same logic helps engineers optimize assembly lines or programmers write more efficient, modular code by identifying common patterns and factoring them out.

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