Problem 86 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{7 n^{2}}{m-n}+\frac{3 m}{n-m}-\frac{3 m^{2}-n}{m^{2}-2 m n+n^{2}}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{7n^2m - 7n^3 - 6m^2 + 3mn + n}{(m-n)^2} \).

Step 1: Recognize the Expressions

First, note the terms: \( \frac{7n^2}{m-n} \), \( \frac{3m}{n-m} \), and \( \frac{3m^2-n}{m^2-2mn+n^2} \). Recognize that \( n-m = -(m-n) \), allowing us to rewrite \( \frac{3m}{n-m} \) as \( -\frac{3m}{m-n} \).

Step 2: Simplify the Denominator of the Third Term

Notice that \( m^2 - 2mn + n^2 \) can be rewritten as \((m-n)^2\). Thus, \( \frac{3m^2-n}{m^2-2mn+n^2} = \frac{3m^2-n}{(m-n)^2} \).

Step 3: Combine Fractions with a Common Denominator

The common denominator for all terms is \((m-n)^2\). Rewrite the fractions with this common denominator:1. \( \frac{7n^2}{m-n} = \frac{7n^2(m-n)}{(m-n)^2} \).2. \( -\frac{3m}{m-n} = \frac{-3m(m-n)}{(m-n)^2} \).3. \( \frac{3m^2-n}{(m-n)^2} \) remains the same.

Step 4: Apply Algebraic Operations

Combine these expressions:\[\frac{7n^2(m-n) - 3m(m-n) - (3m^2-n)}{(m-n)^2}\].Simplify the numerator:\( 7n^2(m-n) - 3m(m-n) = 7n^2m - 7n^3 - 3m^2 + 3mn \).Combine with the third term:\( 7n^2m - 7n^3 - 3m^2 + 3mn - 3m^2 + n = 7n^2m - 7n^3 - 6m^2 + 3mn + n \).

Step 5: Simplify the Result

The full expression is now:\[\frac{7n^2m - 7n^3 - 6m^2 + 3mn + n}{(m-n)^2}\].Check if further simplification is possible by factoring the numerator, but since no straightforward factoring is visible, this is the simplified version.

Key Concepts

Fraction Operations

When dealing with algebraic expressions that involve fractions, understanding fraction operations is crucial. Let's break down the basic operations: addition, subtraction, multiplication, and division of fractions.

• Adding and Subtracting Fractions: This operation requires a common denominator. Once you've identified a common denominator, you can combine the numerators while keeping the denominator the same.
• Multiplying Fractions: Simply multiply the numerators together and the denominators together. This operation is straightforward, and there's no need to find a common denominator.
• Dividing Fractions: Flip the second fraction (take its reciprocal) and then multiply it by the first fraction.

These basic operations are essential for manipulating algebraic expressions further. For instance, in our original exercise, fractions are combined by finding a common denominator before adding or subtracting them.

Common Denominators

Finding a common denominator is a fundamental step in solving problems involving the addition or subtraction of fractions. A common denominator allows you to manipulate fractions as if they were whole numbers.

• Identifying the Common Denominator: Assess each term's denominator and determine the least common multiple (LCM) or use a convenient expression that contains all factors, as needed.
• Adjusting Each Term: Once the common denominator is established, adjust each term by multiplying both the numerator and the denominator by necessary factors to reach the common denominator.

In the solution provided, you see the terms like \( \frac{7n^2}{m-n} \) adjusted to fit the common denominator \((m-n)^2\), enabling smooth operation across the entire expression. This adjustment is critical for ensuring the fractions can be summed or subtracted effectively.

Simplifying Expressions

Simplifying expressions in algebra is the process of making an algebraic expression as concise as possible without changing its value. This often involves reducing terms, combining like terms, and factoring when possible.

• Reduce Complexity: Break down complex terms into simpler components, possibly through factoring or expanding polynomial expressions.
• Combine Like Terms: Identify terms that have the same variable component and combine them to condense the expression.
• Final Review for Further Simplification: After combining like terms, re-evaluate the expression to check if additional factoring or reduction of terms is possible.

In the provided solution, the aim was to consolidate all terms over a single common denominator, simplify the resultant expression by combining like terms, and check for any further reduction possibilities. In this exercise, the complete simplification results in a final expression that is as concise as it can be given the initial conditions.