Problem 96 - Elementary Algebra

Question

(a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Factor \(6 x y-2 x p+3 y-p\) by grouping. Incorrect Answer: \(6 x y-2 x p+3 y-p\) $$ \begin{aligned} &=2 x(3 y-p)+(3 y-p) \\ &=2 x(3 y-p) \end{aligned} $$

Step-by-Step Solution

Answer

The mistake was in the grouping of terms. Correctly factor as \((3y - p)(2x + 1)\).

Step 1: Identify the Given Problem and Incorrect Solution

The problem is to factor the expression \(6xy - 2xp + 3y - p\) by grouping. The incorrect solution provided has steps where a mistake needs to be identified.

Step 2: Identify the Mistake in the Incorrect Solution

Review the incorrect solution which groups the terms as \(2x(3y - p) + (3y - p)\) and then factors as \((2x + 1)(3y - p)\). The mistake is the incorrect factor grouping and combination.

Step 3: Correctly Group the Terms

Correctly group the terms in pairs: \(6xy - 2xp + 3y - p\). Group them as: \((6xy - 2xp) + (3y - p)\).

Step 4: Factor Out the Common Terms in Each Group

For the first group \(6xy - 2xp\), factor out the common term \(2x\): \(2x(3y - p)\). For the second group \(3y - p\), there is no common factor, so it remains as \(3y - p\).

Step 5: Combine the Groups

After factoring out, the expression should be \(2x(3y - p) + 1(3y - p)\). Notice that \(3y - p\) is a common factor.

Step 6: Factor Out the Common Term

Factor out the common binomial factor \(3y - p\): \((3y - p)(2x + 1)\). Therefore, the correctly factored expression is \((3y - p)(2x + 1)\).

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used in algebra to factor certain polynomials. It simplifies expressions by grouping terms into pairs, factoring common terms from each group, and combining these groups. This technique is especially handy when dealing with four-term polynomials.
In our example, we started with the expression: \(6xy - 2xp + 3y - p\).
Here’s the step-by-step approach:

• Group the terms: Divide the terms into two groups: \((6xy - 2xp) + (3y - p)\).
  
• Factor out the common terms: For the first group, factor out \(2x\), giving us \(2x(3y - p)\). The second group doesn’t have a common factor, so it stays \(3y - p\).
  
• Recognize the common binomial factor: We see that \(3y - p\) appears in both groups. We factor this common binomial factor out: \((3y - p)(2x + 1)\).
  

With practice, factoring by grouping becomes a powerful tool for simplifying complex algebraic expressions.

Common Factors

In algebra, a common factor is a term that divides two or more terms without leaving a remainder. Identifying common factors is a crucial step in simplifying and factoring expressions.
In our example, the expression \(6xy - 2xp + 3y - p\) was first split into two groups: \((6xy - 2xp)\) and \((3y - p)\).
When examining the first group, we noticed that both terms, \(6xy\) and \(2xp\), can be divided by \(2x\).
This common factor is then factored out, simplifying the group to \(2x(3y - p)\).
In some cases, you might not find common factors among terms in both groups. As in the second group, \(3y - p\), there was no factor to bring out.
Understanding common factors helps in recognizing patterns and simplifying algebraic expressions efficiently, an invaluable skill in higher levels of math.

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and operators (such as +, -, *, and /). Algebraic expressions can range from simple to highly complex.
In the expression \(6xy - 2xp + 3y - p\), we have:

• Constants: Numbers without variables, though implicitly combined with variables they act as coefficients (like 6, 2, 3, etc.).
  
• Variables: Symbols that represent unspecified numbers (such as \(x, y, p\)).
  
• Terms: Individual parts of an expression separated by + or - (like \(6xy\), \(-2xp\), \(3y\), \(-p\)).
  
• Operations: Mathematical processes applied between terms (addition and subtraction in this case).
  

To factor an algebraic expression effectively, you need to understand the roles of these components. The procedure often involves rearranging terms, identifying patterns, and combining like terms.
This foundational understanding is vital for tackling more challenging problems in algebra and beyond.