Problem 54 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{n^{2}-4 n-21}{n} \div \frac{3-n}{n}\)

Step-by-Step Solution

Answer

Simplified expression: \(-\frac{(n-7)(n+3)}{n-3}\).

Step 1: Understand the Question

We need to divide the expression \( \frac{n^{2}-4n-21}{n} \) by \( \frac{3-n}{n} \). This involves performing a division of two algebraic fractions.

Step 2: Rewrite Division as Multiplication

To divide by a fraction, we multiply by its reciprocal. Therefore, \( \frac{n^{2}-4n-21}{n} \div \frac{3-n}{n} \) becomes \( \frac{n^{2}-4n-21}{n} \times \frac{n}{3-n} \).

Step 3: Simplify the Expression

Let's simplify the expression \( \frac{n^{2}-4n-21}{n} \times \frac{n}{3-n} \). First, notice that the \(n\) terms in the numerator and denominator will cancel each other out. This simplifies the expression to \( \frac{n(n-4)-21}{3-n} \).

Step 4: Factor the Numerator

The numerator \( n^2 - 4n - 21 \) can be factored. The expression factors as \( (n-7)(n+3) \) because \( -7 \times 3 = -21 \) and \( -7 + 3 = -4 \). The expression simplifies to \( \frac{(n-7)(n+3)}{3-n} \).

Step 5: Simplify Using Negative Transformation

Recognize that \( 3-n = -(n-3) \). Thus we can rewrite the expression as \( -\frac{(n-7)(n+3)}{n-3} \).

Step 6: Conclusion

The simplified expression is \(-\frac{(n-7)(n+3)}{n-3}\). It cannot be simplified further because there are no common factors between the numerator and the denominator.

Key Concepts

Division of Fractions

When dividing algebraic fractions, it is crucial to understand that division is essentially the same as multiplying by the reciprocal. Imagine you have the expression \( \frac{a}{b} \div \frac{c}{d} \). To divide these fractions, you would multiply the first fraction by the reciprocal of the second. This means it becomes \( \frac{a}{b} \times \frac{d}{c} \).

• Reciprocal: The reciprocal of a fraction \( \frac{c}{d} \) is \( \frac{d}{c} \).
• Simplifying: After rewriting the division as multiplication, always check if the numerators and denominators have common factors you can simplify.

In our exercise, we started with \( \frac{n^2 - 4n - 21}{n} \div \frac{3-n}{n} \). We turned this into a multiplication: \( \frac{n^2 - 4n - 21}{n} \times \frac{n}{3-n} \). This technique allows us to work with the expression more flexibly and eventually simplify it further.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing their value. In algebraic fractions, this usually involves canceling out common factors in the numerator and the denominator.

• Cancel Common Factors: Check both the numerator and denominator for any terms that can be canceled.
• Simplify Step-by-Step: Follow each simplification step carefully, ensuring that equivalent expressions don't become altered inadvertently.

For instance, in the expression \( \frac{n^2 - 4n - 21}{n} \times \frac{n}{3-n} \), we can cancel the \( n \) terms because one appears in both the numerator of one fraction and the denominator of another. This simplification results in \( \frac{n(n-4) - 21}{3-n} \), making it easier to manage further factoring.

Factoring Polynomials

Factoring polynomials means breaking down a complex expression into simpler, multiplied expressions. It is like pulling apart a tangled knot into its individual strings.

• Look for Common Patterns: Often, polynomials factor into products of binomials or other polynomials.
• Confirm by Multiplication: Always check that your factored expressions multiply back to the original polynomial; this ensures accuracy.

In the problem we faced, we took \( n^2 - 4n - 21 \) and factored it into \( (n-7)(n+3) \). This is because \( -7 \times 3 = -21 \) and \( -7 + 3 = -4 \). Recognizing such patterns helps simplify the expression, bringing us closer to the solution. Factoring is not only essential for simplifying fractions but also plays a crucial role in solving equations.