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Chapter 5: Problem 55

Write an equivalent expression by factoring. $$2 x y+3 x-x^{2} y-6$$

Short Answer

Expert verified
(2 - x)(xy - 3)

Step by step solution

01

Group the terms

First, group the terms to make factoring easier. The original expression is: \[ 2xy + 3x - x^2 y - 6 \]Group the terms as follows:\[ (2xy - x^2 y) + (3x - 6) \]
02

Factor out the greatest common factor (GCF) from each group

From the first group \( (2xy - x^2 y) \), factor out the GCF, which is \( xy \):\[ xy(2 - x) \]From the second group \( (3x - 6) \), factor out the GCF, which is \( 3 \):\[ 3(x - 2) \]
03

Combine factored groups

Rewrite the expression combining the factored groups:\[ xy(2 - x) + 3(x - 2) \]
04

Rearrange terms inside the second factor

Notice that \(2 - x\) and \(x - 2\) are very similar, differing only by a factor of -1. Factor \(-1\) outside the second term to match the first:\[ xy(2 - x) + 3(-1)(2 - x) \]Simplify:\[ xy(2 - x) - 3(2 - x) \]
05

Factor out the common binomial

Now factor out the common binomial \(2 - x\):\[ (2 - x)(xy - 3) \]

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Key Concepts

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

Grouping Terms
When tackling polynomial expressions, grouping terms can simplify the process. Given our original expression, [2xy + 3x - x^2 y - 6], the idea is to group terms that share common factors. This grouping helps us to factor out the greatest common factor more easily. For example, we can group the expression as follows: [(2xy - x^2 y) + (3x - 6)]. Notice how we put the terms with similar variables together. Grouping terms is like arranging puzzle pieces to see patterns that make factoring straightforward.
Greatest Common Factor (GCF)
The greatest common factor (GCF) is the largest factor that divides two or more terms. Identifying the GCF in each group is crucial for simplifying polynomial expressions. Let鈥檚 look at our expression [2xy - x^2 y]. Both terms have the variable [xy], so we factor out [xy]: [xy(2 - x)]. Likewise, for the group [3x - 6], the common factor is [3], giving us: [3(x - 2)]. Finding the GCF speeds up the factoring process and ensures accuracy.
Factoring Binomials
Factoring binomials involves expressing a polynomial as a product of simpler polynomials. Here, after grouping like terms and factoring out the GCF, we rewrite and simplify: [xy(2 - x) + 3(x - 2)]. Notice that [2 - x] and [x - 2] differ by a factor of [-1]. Rewriting [3(x - 2)] to match the first binomial gives us: [xy(2 - x) - 3(2 - x)]. Now, [2 - x] is a common binomial factor. Factoring it out, we get: [(2 - x)(xy - 3)]. This process simplifies the expression to its factored form.

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Most popular questions from this chapter

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