Problem 20 - Algebra 2 and Trigonometry

Question

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{w^{2}-w}{5 w} \div \frac{w^{2}-1}{5} $$

Step-by-Step Solution

Answer

\( \frac{1}{w+1} \), undefined for \( w = -1, 0, 1 \).

Step 1: Rewrite Division as Multiplication

To divide fractions, we multiply by the reciprocal of the second fraction. So, rewrite the given expression as: \( \frac{w^2 - w}{5w} \times \frac{5}{w^2 - 1} \).

Step 2: Factor the Expressions

Factor the numerators and denominators where possible. For \( w^2 - w \), factor out \( w \): \( w(w-1) \). For \( w^2 - 1 \), recognize it as a difference of squares: \( (w+1)(w-1) \).

Step 3: Simplify Across Multiplication

Now substitute the factored terms back: \( \frac{w(w-1)}{5w} \times \frac{5}{(w+1)(w-1)} \). Cancel common terms across the numerator and the denominator. In this case, \( (w-1) \) in the numerator and denominator are common, and \( 5 \) also cancels out, resulting in: \( \frac{w}{w(w+1)} \).

Step 4: Cancel Common Terms

Further simplify by canceling \( w \) in the numerator and denominator: \( \frac{1}{w+1} \).

Step 5: Identify Undefined Values

Expressions are undefined when the denominator is zero. For the original expression: \( 5w(w+1) = 0 \) if \( w = 0 \) or \( w = -1 \). Also, note \( w - 1 = 0 \) when \( w = 1 \) in the original fractions. Hence, the variables for which the fractions are not defined are \( w = 0, 1, -1 \).

Key Concepts

Factoring Polynomials

When working with polynomial expressions, factoring is a crucial step in simplifying them, especially in rational expressions. If we look at the expression \( w^2 - w \), factoring involves finding common factors that simplify the form. Here, we notice that both terms, \( w^2 \) and \( -w \), share a common factor: \( w \). So, we factor \( w \) out of the expression to get \( w(w - 1) \).

Another common pattern in factoring is recognizing special structures such as the difference of squares, which occurs when the expression is in the form of \( a^2 - b^2 \). For \( w^2 - 1 \), this can be expressed as \((w + 1)(w - 1)\), because\( 1 \) can be rewritten as \( 1^2 \). This insight into the structure makes simplifying complex expressions much more manageable.

• Always look for common factors first.
• Recognize special structures like the difference of squares.
• Factoring simplifies expressions and helps identify common terms for cancellation in division.

Undefined Values in Expressions

Expressions become undefined at specific points that can cause the denominator to be zero, making division impossible. It is essential to identify these critical values. For the expression \( \frac{w(w-1)}{5w} \times \frac{5}{(w+1)(w-1)} \), we explore where the denominators are zero. In the step-by-step solution, this is noted as when:\( w = 0 \), \( w = 1 \), or \( w = -1 \).

Finding these values requires setting the entire denominator equal to zero and solving for \( w \). Let’s break it down:

• \( 5w(w+1) = 0 \) implies \( w = 0 \) or \( w+1 = 0 \), leading to \( w = -1 \).
• The factor \( w - 1 = 0 \) gives \( w = 1 \).

It's crucial to identify these points as they ensure the expressions are mathematically valid, safeguarding against division by zero. Always perform these checks when dealing with rational expressions to ensure the final expression is valid over its domain.

Division of Fractions

Dividing fractions might seem tricky at first, but with a simple rule, it becomes manageable. The key process is to multiply by the reciprocal of the divisor. For example, dividing \( \frac{w^2 - w}{5w} \) by \( \frac{w^2 - 1}{5} \) is the same as multiplying \( \frac{w^2 - w}{5w} \) by the reciprocal of \( \frac{w^2 - 1}{5} \), which is \( \frac{5}{w^2 - 1} \).

So, change \( \div \) to a multiplication, and flip the second fraction. This switch makes simplifying so much easier once we've factored the polynomials. After the conversion, simplify by canceling out equal terms from the numerator and denominator, which reduces complexity. Remember to:

• Rewrite division as multiplication by the reciprocal.
• Factor polynomials before multiplying.
• Cancel common factors wisely.

By mastering this method, complex expressions shrink significantly, leaving you with a simpler outcome that is often easier to interpret or to use further.