Problem 88 - Algebra for College Students

Question

Perform the indicated operations. Variables in exponents represent integers. $$\frac{w^{2 b}+2 w^{b}-8}{w^{2 b}+3 w^{b}-4} \div \frac{w^{2 b}-w^{b}-2}{w^{2 b}-1}$$

Step-by-Step Solution

Answer

1

Step 1: Factor the Numerators and Denominators

First, factor the numerators and denominators of each fraction. For the numerator of the first fraction, \[w^{2b} + 2w^b - 8 = (w^b + 4)(w^b - 2)\]For the denominator of the first fraction, \[w^{2b} + 3w^b - 4 = (w^b + 4)(w^b - 1)\]For the numerator of the second fraction, \[w^{2b} - w^b - 2 = (w^b - 2)(w^b + 1)\]For the denominator of the second fraction, \[w^{2b} - 1 = (w^b - 1)(w^b + 1)\]

Step 2: Rewrite the Division as Multiplication

Rewrite the given expression by converting the division into multiplication by the reciprocal. \[\frac{(w^b + 4)(w^b - 2)}{(w^b + 4)(w^b - 1)} \div \frac{(w^b - 2)(w^b + 1)}{(w^b - 1)(w^b + 1)} = \frac{(w^b + 4)(w^b - 2)}{(w^b + 4)(w^b - 1)} \times \frac{(w^b - 1)(w^b + 1)}{(w^b - 2)(w^b + 1)}\]

Step 3: Cancel Common Factors

Identify and cancel out the common factors in the numerator and denominator. In this case, \((w^b + 4), (w^b - 2), (w^b - 1), \text{and } (w^b + 1)\) are the common factors that can be canceled. The remaining terms are: \[\frac{1}{1} = 1\]

Key Concepts

Factoring Polynomials

Polynomials can often be broken down into simpler factors. This process is called factoring, and it’s crucial in simplifying complex algebraic expressions. In the original problem, both the numerators and denominators of the given fractions are factored. For example, the polynomial \(w^{2b} + 2w^b - 8\) factors to \((w^b + 4)(w^b - 2)\). Factoring involves finding two binomials that, when multiplied together, give the original polynomial. Here’s a general approach to factor polynomials:

• Look for common factors.
• Use patterns like the difference of squares \(a^2 - b^2 = (a - b)(a + b)\).
• Consider trial and error for quadratic trinomials.

Factoring helps reveal simpler expressions, which are key for further operations like reducing fractions.

Simplifying Rational Expressions

Rational expressions are fractions involving polynomials in the numerators and denominators. Simplifying them requires factoring and canceling out common factors. In the original problem, the expressions are simplified by rewriting and then canceling common factors:

• Identify common factors in the numerator and denominator.
• Cancel out these common factors to simplify the expression.

For example, in the expression \(\frac{(w^b + 4)(w^b - 2)}{(w^b + 4)(w^b - 1)} \times \frac{(w^b - 1)(w^b + 1)}{(w^b - 2)(w^b + 1)}\), common factors like \(w^b + 4\) and \(w^b - 2\) are canceled, simplifying the expression to \(\frac{1}{1} = 1\). This simplification process reduces the complexity of the overall expression, making it more manageable and easier to work with.

Division of Rational Expressions

Dividing rational expressions involves inverting the second fraction and switching to multiplication. This operation is similar to fraction division:

• Take the reciprocal (invert) the second fraction.
• Rewrite the division as a multiplication problem.
• Factor and simplify as done with multiplication.

In the problem, the given division \(\frac{(w^b + 4)(w^b - 2)}{(w^b + 4)(w^b - 1)} \div \frac{(w^b - 2)(w^b + 1)}{(w^b - 1)(w^b + 1)}\) is inverted to multiply by the reciprocal: \(\frac{(w^b + 4)(w^b - 2)}{(w^b + 4)(w^b - 1)} \times \frac{(w^b - 1)(w^b + 1)}{(w^b - 2)(w^b + 1)}\). After this, it becomes easier to factor and cancel out common terms, leading to the simplified result.