Problem 89 - Algebra for College Students

Question

Perform the indicated operations. Variables in exponents represent integers. $$\frac{m^{k} v^{k}+3 v^{k}-2 m^{k}-6}{m^{2 k}-9} \cdot \frac{m^{2 k}-2 m^{k}-3}{v^{k} m^{k}-2 m^{k}+2 v^{k}-4}$$

Step-by-Step Solution

Answer

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

Step 1 - Factor the Numerators

First, factor the numerators of both fractions: 1. For the first numerator: \(m^{k} v^{k} + 3 v^{k} - 2 m^{k} - 6\). Grouping terms gives: \((m^{k} v^{k} - 2 m^{k}) + (3 v^{k} - 6)\). We can factor out common factors: \(m^{k}(v^{k}-2) + 3(v^{k}-2)\). Now, factor out \((v^{k}-2)\): \((v^{k}-2)(m^{k}+3)\). 2. For the second numerator: \(m^{2 k} - 2 m^{k} - 3\). Factor to obtain: \((m^{k}-3)(m^{k}+1)\).

Step 2 - Factor the Denominators

Next, factor the denominators of both fractions: 1. For the first denominator: \(m^{2 k} - 9\). Recognize this as a difference of squares: \((m^{k} - 3)(m^{k} + 3)\). 2. For the second denominator: \(v^{k} m^{k} - 2 m^{k} + 2 v^{k} - 4\). Group the terms: \((v^{k} m^{k} + 2 v^{k}) - (2 m^{k} + 4)\). Factor common factors out of each group: \(v^{k}(m^{k} + 2) - 2(m^{k} + 2)\). Factor out the common term \((m^{k} + 2)\): \((m^{k}+2)(v^{k}-2)\).

Step 3 - Simplify the Expression

Rewrite the expression with the factored forms: \[\frac{(v^{k}-2)(m^{k}+3)}{(m^{k}-3)(m^{k}+3)} \cdot \frac{(m^{k}-3)(m^{k}+1)}{(m^{k}+2)(v^{k}-2)}\]. Cancel out common factors in the numerator and denominator: \(v^{k}-2\) and \(m^{k}-3\) and \(m^{k}+3\) are common factors. This simplifies to: \frac{m^{k}+1}{m^{k}+2}\.

Key Concepts

Polynomial Expressions

A polynomial expression is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. For example, in the exercise, we have expressions like \(m^{k} v^{k} + 3 v^{k} - 2 m^{k} - 6\). Polynomials can include constants, variables, and exponents that are non-negative integers.
Understanding polynomial expressions is crucial, as they are the building blocks for more complex algebraic concepts.

• Each term in a polynomial is called a monomial.
• Polynomial degrees are determined by the highest exponent present in the expression.
• Operations with polynomials include addition, subtraction, multiplication, and division.

In the given problem, polynomial expressions appear within both the numerators and denominators of the fractions.

Factoring Trinomials

Factoring trinomials involves breaking down a polynomial with three terms into simpler binomials. In our case, looking at the second numerator, \(m^{2 k} - 2 m^{k} - 3\), we transform this into \(m^{k} - 3)(m^{k} + 1)\).
This process is essential for simplifying expressions and solving polynomial equations.

• Identify the trinomial structure \(ax^2+bx+c\).
• Look for two numbers that multiply to \(ac\) and add to \(b\).
• Rewrite the middle term using these numbers, then factor by grouping.

Practice ensures proficiency, improving your ability to handle algebraic problems efficiently.

Simplification of Rational Expressions

Simplification of rational expressions entails reducing fractions involving polynomials. For instance, after factoring as seen above, we can rewrite the original complex expression into simpler terms. This involves canceling out common factors from the numerator and denominator.
Steps for simplification include:

• Factor both the numerator and the denominator.
• Identify and cancel out common factors.
• Rewrite the simplified form.

In our exercise, the simplified form resulted in \(\frac{m^{k}+1}{m^{k}+2}\). Simplification is crucial for making complex algebraic expressions more manageable.

Difference of Squares

The difference of squares is a specific type of polynomial expression that can be factored neatly. It follows the pattern \(a^2 - b^2 = (a - b)(a + b)\). This formula was applied to the first denominator, \(m^{2 k} - 9\), recognizing it as \(m^{2 k} - 3^2 = (m^{k} - 3)(m^{k} + 3)\).
Recognizing this pattern simplifies algebraic manipulation.

• A square term is any expression where an exponent is 2, e.g., \(9 = 3^2\).
• Look for subtraction between two squared terms.
• Factor using the formula provided.

Mastering the difference of squares increases your ability to quickly factor and simplify expressions.