Problem 83 - Algebra for College Students

Question

Perform the indicated operations. $$\frac{k^{2}+2 k m+m^{2}}{k^{2}-2 k m+m^{2}} \cdot \frac{m^{2}+3 m-m k-3 k}{m^{2}+m k+3 m+3 k}$$

Step-by-Step Solution

Answer

\( \frac{-(k + m)}{k - m} \)

Step 1: Factor the Numerator of the First Fraction

Identify if the numerator of the first fraction can be factored. The numerator is: \(k^{2} + 2km + m^{2}\) This is a perfect square trinomial and can be factored as: \((k + m)^{2}\)

Step 2: Factor the Denominator of the First Fraction

Identify if the denominator of the first fraction can be factored. The denominator is: \(k^{2} - 2km + m^{2}\) This is a perfect square trinomial and can be factored as: \((k - m)^{2}\)

Step 3: Factor the Numerator of the Second Fraction

Factor the polynomial in the numerator of the second fraction. The numerator is: \(m^{2} + 3m - mk - 3k\) Group and factor each part: \(m(m + 3) - k(m + 3)\) Factor out \((m + 3)\): \((m + 3)(m - k)\)

Step 4: Factor the Denominator of the Second Fraction

Factor the polynomial in the denominator of the second fraction. The denominator is: \(m^{2} + mk + 3m + 3k\) Group and factor each part: \(m(m + k) + 3(m + k)\) Factor out \((m + k)\): \((m + k)(m + 3)\)

Step 5: Substitute Factored Forms and Simplify

Substitute the factored forms into the fractions and simplify: \[\frac{(k + m)^{2}}{(k - m)^{2}} \cdot \frac{(m + 3)(m - k)}{(m + k)(m + 3)}\] Notice that \((m + 3)\) cancels out. The expression simplifies to: \[\frac{(k + m)^{2}(m - k)}{(k - m)^{2}(m + k)}\]

Step 6: Further Simplify the Expression

Rewrite \((m - k)\) as \(-(k - m)\): \[\frac{(k + m)^{2}(-1)(k - m)}{(k - m)^{2}(m + k)}\] Cancel \((k - m)\) from the numerator and denominator, this leaves: \[\frac{-(k + m)}{k - m}\]

Key Concepts

Factoring Trinomials

Factoring trinomials is an important concept in algebra. It involves rewriting a trinomial as a product of two binomials. To factor effectively, you need to recognize patterns. In this example, the trinomials in the first fraction's numerator and denominator are both perfect squares:

- \( k^2 + 2km + m^2 \) becomes \( (k + m)^2 \)
- \( k^2 - 2km + m^2 \) becomes \( (k - m)^2 \)

Recognizing perfect square trinomials quickly can simplify a lot of algebraic problems and make further operations easier.

Simplifying Fractions

Simplifying fractions is crucial when dealing with rational expressions. The key steps involve factoring both the numerator and the denominator and canceling out common factors.

In our example, after factoring, we have:

\[ \frac{(k + m)^2}{(k - m)^2} \times \frac{(m + 3)(m - k)}{(m + k)(m + 3)} \]

Notice that \( (m + 3) \) appears in both the numerator and the denominator of the second fraction. We can cancel it out, simplifying the expression to:

\[ \frac{(k + m)^2(m - k)}{(k - m)^2(m + k)} \]

This step of simplifying fractions is essential to make the problem more manageable.

Polynomial Operations

Polynomial operations, such as addition, subtraction, multiplication, and factorization, are fundamental in algebra. For simplifying rational expressions, multiplication and factorization are key.

In this exercise, we multiplied two fractions after factoring all polynomials involved. We saw how factors like \( m + 3 \) can cancel each other out when present in both numerator and denominator. Also, rewriting expressions helps in further simplification. For example, \( (m - k) \) can be rewritten as \( -(k - m) \), which aids in reducing the complexity of the fraction:

\[ \frac{(k + m)^2(-1)(k - m)}{(k - m)^2(m + k)} \]

Algebra Solutions

Algebra solutions require a systematic approach. Start by factoring, simplifying, performing operations, and then simplifying further if possible. In combining fractions and simplifying rational expressions, following these steps diligently ensures accurate results.

For this particular exercise, the systematic approach was:

• Factoring trinomials (perfect squares and grouping)
• Simplifying the fractions by canceling out common factors
• Rewriting expressions to facilitate further simplification

This led to our final simplified answer:

\[ \frac{-(k + m)}{k - m} \]

We used consistent steps and logical problem-solving methods. This ensures clarity and helps in understanding the broader concepts of algebra and rational expressions.