Problem 80 - Algebra and Trigonometry

Question

Factor the expression completely. $$ n(x-y)+(n-1)(y-x) $$

Step-by-Step Solution

Answer

The expression factors to \(x-y\).

Step 1: Arrange the terms

Notice that the terms \((x-y)\) and \((y-x)\) are similar but in opposite order. We know \((y-x) = -(x-y)\), so let's rewrite the expression as: \[ n(x-y) - (n-1)(x-y) \].

Step 2: Factor by grouping

Since both terms now have a common factor of \((x-y)\), we can factor \((x-y)\) out of the expression. This gives us: \[ (x-y)(n - (n-1)) \].

Step 3: Simplify the expression

Now, simplify the expression inside the parentheses. We have \(n - (n-1)\), which simplifies to:\[ n - n + 1 = 1 \].Thus, the expression becomes \((x-y) \cdot 1\).

Step 4: Final expression

Since multiplying by 1 does not change the value, the completely factored expression is:\[ x - y \].

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. They can range from simple expressions, like \(x+y\), to more complex ones with multiple terms and operations. Each part of an algebraic expression, separated by plus or minus signs, is called a 'term.' For instance, in the expression \(n(x-y)+(n-1)(y-x)\), there are two terms: \(n(x-y)\) and \((n-1)(y-x)\).

Understanding the structure of algebraic expressions is crucial for simplifying and factoring them. The key is to identify variables (\(x,y\)), coefficients (numbers like \(n\) or \(n-1\)), and how they are combined. Expressing relationships between variables makes it easier to manipulate and solve expressions efficiently.

Common Factors

Finding common factors is essential in factoring expressions. A 'common factor' is an element shared by all terms of the expression. In an expression like \(n(x-y)-(n-1)(x-y)\), notice that \((x-y)\) appears in both terms.

To identify a common factor:

• Look for terms that repeat in different parts of the expression.
• Sometimes the order or sign might change, as in \(y-x\) and \(x-y\), which only differ by a negative sign.
• Rewrite or reorganize terms if needed to make common factors more visible.

Recognizing common factors allows for factoring processes like grouping to simplify the expression.

Simplification

Simplification involves rewriting an expression in its most basic or compact form. This step often includes combining like terms and reducing expressions to their simplest state.

Consider \(n - (n-1)\) from the expression \((x-y)(n - (n-1))\)\. It simplifies to \(n - n + 1 = 1\), which can make your calculations or equations much less burdensome.

Here are some tips for simplification:

• Always look for similar terms that can be combined.
• Use basic arithmetic operations like addition and subtraction to condense expressions.
• Remember that any expression multiplied by 1 stays unchanged, simplifying the visual complexity but not numerical value.

Simplified expressions are easier to handle and can lead to faster problem-solving.

Factoring by Grouping

Factoring by grouping is a technique used when an expression has four or more terms. It involves organizing and grouping terms to extract common factors.

When you have an expression like \(n(x-y)-(n-1)(x-y)\), note how \((x-y)\) is a common factor. Factoring by grouping involves:

• Rearranging terms if necessary, to reveal common factors more clearly.
• Pulling out the common factor from each group.
• Ensuring any differences in expression signs are balanced, using negative coefficients if needed.

By grouping, \((x-y)\) could be factored out, leaving a simpler expression inside the parentheses. Ultimately, the result is an elegant and manageable expression like \(x-y\), easier to interpret and use.