Problem 51 - Elementary and Intermediate Algebra

Question

Simplify. See Example \(5 .\) $$ \frac{m^{2}-2 m n+n^{2}}{7 m^{2}-7 n^{2}} $$

Step-by-Step Solution

Answer

\( \frac{m-n}{7(m+n)} \)

Step 1: Factor the Numerator

The numerator is \( m^2 - 2mn + n^2 \). This is a perfect square trinomial and can be factored as \((m-n)^2\) using the formula \((a-b)^2 = a^2 - 2ab + b^2\).

Step 2: Factor the Denominator

The denominator is \( 7m^2 - 7n^2 \). This expression is a difference of squares and can be factored using the formula \( a^2 - b^2 = (a-b)(a+b) \). Factor out the common factor \( 7 \) first to get \( 7(m^2-n^2) \), and then factor \( m^2 - n^2 \) as \((m-n)(m+n)\). Thus, the complete factorization is \( 7(m-n)(m+n) \).

Step 3: Cancel Common Factors

Now that we have \( \frac{(m-n)^2}{7(m-n)(m+n)} \), we see a common factor of \( m-n \) in both the numerator and the denominator. Cancel out \( m-n \) from both, which simplifies the expression to \( \frac{m-n}{7(m+n)} \).

Step 4: Write the Final Simplified Expression

After cancellation, the final simplified expression is \( \frac{m-n}{7(m+n)} \). Ensure that \( m eq n \) to avoid division by zero.

Key Concepts

Factoring

Factoring in algebra is a process used to simplify expressions or solve equations. It involves expressing a polynomial as a product of its factors, which are simpler expressions. Imagine tearing apart a complex object into simpler building blocks.

When you factor, you're essentially looking for two or more expressions that multiply together to get back to the original expression. For instance, in the original exercise, the numerator is a perfect square trinomial and can be expressed as

• \((m^2 - 2mn + n^2)\).
• This trinomial can be rewritten as \((m-n)^2\).

Factoring is particularly useful for recognizing patterns, which makes solving algebraic expressions more efficient.

Simplification

Simplification makes expressions easier to work with by reducing them to their simplest form. It involves many strategies, mainly including factoring and canceling common factors.

In the exercise, simplification is achieved through the cancellation of the common term

• \((m-n)\) in both the numerator and the denominator.
• This reduces \(\frac{(m-n)^2}{7(m-n)(m+n)}\) to \(\frac{m-n}{7(m+n)}\).

Simplification helps in reducing error when solving equations and provides clarity in mathematical reasoning.

Perfect Square Trinomials

Perfect square trinomials are specific types of algebraic expressions that result from squaring a binomial. Recognizing these trinomials is key to factoring effectively.

These trinomials follow a recognizable pattern: \(a^2 - 2ab + b^2\). For instance,

• \((m^2 - 2mn + n^2)\) is a perfect square trinomial.
• By using the formula \((a-b)^2 = a^2 - 2ab + b^2\), it can be factored as \((m-n)^2\).

Spotting perfect square trinomials can simplify many mathematical problems, making this concept vital in algebra.

Difference of Squares

The difference of squares is a powerful method for factoring expressions of the form \(a^2 - b^2\). Such expressions can be factored into the product of two conjugates: \((a-b)(a+b)\).

In the denominator of the exercise, you have the expression \(7(m^2 - n^2)\), which is a multiple of two squares. It's factored as follows:

• Factor out the \(7\) to get \(7(m^2-n^2)\).
• Then, recognize \(m^2-n^2\) as a difference of squares and factor it into \((m-n)(m+n)\), leading to \(7(m-n)(m+n)\).

This technique helps to break down expressions into simpler, manageable parts, critical in solving equations.