Problem 74 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ n(x-y)+(n-1)(y-x) $$

Step-by-Step Solution

Answer

The expression is factored completely as \((x-y)\).

Step 1: Identify the Expression

We start with the expression \( n(x-y)+(n-1)(y-x) \) which needs to be factored completely.

Step 2: Simplify Elementary Terms

Note that \((y-x) = -(x-y)\). Rewrite the expression as \( n(x-y) - (n-1)(x-y) \) by using this property.

Step 3: Combine Like Terms

Factor out \((x-y)\) from each term: \((x-y)(n - (n-1))\).

Step 4: Simplify Parentheses

Simplify the expression inside the parentheses: \(n - (n-1) = 1\). Thus, the expression becomes \((x-y) \cdot 1\).

Step 5: Final Result

The completely factored form of \( n(x-y)+(n-1)(y-x) \) is simply \((x-y)\).

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions composed of variables and coefficients, involving operations like addition, subtraction, and multiplication. They do not have division by a variable. For example, in the expression \( ax^2 + bx + c \), \(a\), \(b\), and \(c\) are coefficients, while \(x\) is the variable. Courting simplicity, polynomials can be understood as sums of terms, where each term contains a constant multiplied by a variable raised to a non-negative whole number.

Handling polynomial expressions often requires rearranging and simplifying terms. This ensures easier computation and interpretation, particularly in solving equations or factorizing them like in our example. Understanding the structure and degree of a polynomial helps determine the methods for solving or simplifying it. In this exercise, recognizing the polynomial layout allowed us to apply specific properties for factorization.

Distributive Property

The distributive property is a crucial tool in algebra that allows us to multiply a single term across terms in parentheses. It states that for any terms \(a\), \(b\), and \(c\):

• \(a(b + c) = ab + ac\)
• \(a(b - c) = ab - ac\)

In the context of our exercise, we've applied the distributive property to rewrite the expression \( n(x-y) + (n-1)(y-x) \) into \( n(x-y) - (n-1)(x-y) \). This restatement exploits the property \((y-x) = -(x-y)\).

Such manipulations underscore the power of the distributive property in simplifying expressions and identifying common factors. By distributing and rearranging, we can reveal opportunities to factor the expression further, cultivating a cleaner and more manageable form.

Simplification Techniques

Simplification techniques in algebra aim to reduce expressions to their simplest form. This often involves combining like terms, factoring, and using arithmetic identities. In our example, each of these techniques plays a vital role in reaching \((x-y)\) as the fully factored expression.

• Combining Like Terms: By recognizing \((y-x)\) as \(-(x-y)\), we rewrite the expression in a way that highlights common factors.
• Factoring Out: After rewriting, factoring \((x-y)\) from both terms leads to a simpler equation.
• Arithmetic Simplification: Simple subtraction \(n - (n-1)\) simplifies to \(1\), leading to a further reduced expression.

These steps illustrate that simplification often involves creative reframing and use of basic arithmetic operations, not just straightforward calculation. Mastery of these techniques simplifies not only algebraic expressions but also aids in broader mathematical problem solving, ensuring clarity and precision.