Problem 59 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{3 m^{2}-2 m n-n^{2}}{m n-m^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{3m + n}{m}\).

Step 1: Factor the Numerator

First, look at the numerator, which is \(3m^2 - 2mn - n^2\). We will use the technique of factoring quadratics to rewrite this expression in a factored form. By examining the terms, we identify that it can be factored into \((3m + n)(m - n)\).

Step 2: Factor the Denominator

Now, let's factor the denominator, \(mn - m^2\). Notice that we can factor out \(m\) from both terms, resulting in \(m(n - m)\).

Step 3: Simplify the Expression

After factoring both the numerator and the denominator, the expression is \(\frac{(3m + n)(m - n)}{m(n - m)}\). Observe that \((m - n)\) and \((n - m)\) are negatives of each other: \((n - m) = -(m-n)\). So, the terms \((m-n)\) in the numerator and \((n-m)\) in the denominator can cancel, but this introduces a negative sign. This simplifies the expression to \(-\frac{3m + n}{m}\).

Step 4: Final Expression

The result after simplification is \(-\frac{3m + n}{m}\). There are no more factors to cancel, and the expression is now simplified.

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is crucial for simplifying expressions. Factoring is the process of breaking down a polynomial into its simplest parts, which are called factors.
These factors multiply together to give the original expression. Let's focus on the numerator:

• We look for two numbers that multiply to give the coefficient of the quadratic term when combined with the constant term.
• Here, we have the expression \(3m^2 - 2mn - n^2\). To factor this, we identify that it breaks down into \((3m + n)(m - n)\).
• These binomial terms multiply to give back the original quadratic expression.

Factoring is like unraveling a recipe to see its individual ingredients. Once we've successfully factored a quadratic, we can simplify it significantly, making it easier to work with in equations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing expressions to their most basic form. This often makes solving equations much easier. Let's focus on our rational expression here:

• Start by factoring both the numerator and the denominator. We previously factored our expression into \(\frac{(3m+n)(m-n)}{m(n-m)}\).
• Notice that \((m-n)\) in the numerator and \((n-m)\) in the denominator cancel each other out, but introduce a negative sign.
• The expression reduces successfully to \(-\frac{3m + n}{m}\).
• It's always important to check that no further simplification can occur by confirming all terms are in their simplest form.

Simplicity in algebra enables clearer problem-solving, offering greater insight into the problems and solutions before us.

Quadratic Factorization

Quadratic factorization is a method used to simplify quadratic expressions by expressing them as a product of two binomials. For the exercise presented:

• Observe the expression \(3m^2 - 2mn - n^2\).
• We factor it into the form \((3m + n)(m - n)\), where each binomial represents a significant part of the original quadratic expression.
• This technique relies on recognizing patterns or using trial and error to successfully break the quadratic into simpler terms.

Understanding and mastering quadratic factorization can significantly simplify working through various algebraic equations, especially when combined with other mathematical strategies. It leads to easier manipulation and solution of complex expressions, as seen in our exercise.