Problem 61 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{n^{2}-10 n+9}{n-9} \div(n-1)\)

Step-by-Step Solution

Answer

The result of the division is 1.

Step 1: Understand the Problem

We need to divide the rational expression \( \frac{n^{2}-10n+9}{n-9} \) by \( n-1 \). This is equivalent to multiplying \( \frac{n^{2}-10n+9}{n-9} \) by the reciprocal of \( n-1 \), which is \( \frac{1}{n-1} \).

Step 2: Rewrite the Division as Multiplication

Transform the division of \( \frac{n^{2}-10n+9}{n-9} \div (n-1) \) into multiplication:\[ \frac{n^{2}-10n+9}{n-9} \times \frac{1}{n-1} \]

Step 3: Factor the Numerator

Factor the quadratic expression in the numerator, \( n^{2}-10n+9 \). We need to find two numbers that multiply to 9 and add to -10. These numbers are -1 and -9, so:\[ n^{2}-10n+9 = (n-1)(n-9) \]

Step 4: Substitute the Factored Form

Replace the numerator in the expression with its factored form:\[ \frac{(n-1)(n-9)}{n-9} \times \frac{1}{n-1} \]

Step 5: Cancel Common Factors

Cancel the common factors in the expression. The factor \( (n-9) \) can be canceled from the numerator and the denominator, and \( (n-1) \) can be canceled as well:\[ \frac{(n-1)}{1} \times \frac{1}{n-1} = 1 \]

Step 6: Simplify the Expression

After canceling common factors, the simplified expression is 1, as all the factors have been canceled out.

Key Concepts

Rational Expressions

Rational expressions are fractions in which the numerator and/or the denominator is a polynomial. In the given exercise, the expression \( \frac{n^{2}-10n+9}{n-9} \) is a rational expression since both the numerator \( n^{2}-10n+9 \) and the denominator \( n-9 \) are polynomial expressions.

When dealing with rational expressions, it's important to remember that:

• The denominator cannot be zero because division by zero is undefined. So, for \( \frac{n^{2}-10n+9}{n-9} \), \( n \) cannot be 9.
• Rational expressions can often be simplified by factoring and canceling common factors, which makes subsequent calculations easier.

Understanding these key ideas can make working with rational expressions much smoother. Being able to identify them and knowing how to handle division correctly is crucial, as it requires multiplying by the reciprocal as shown in the step-by-step solution.

Factoring Quadratics

Factoring quadratics is a vital skill when working with polynomial expressions, especially in simplifying and solving equations. We are often looking for two numbers that multiply to the constant term and add to the linear coefficient. For the quadratic expression \( n^{2} - 10n + 9 \), we look for numbers that multiply to 9 and add to -10.

In this case, -1 and -9 fit the criteria:

• They multiply to give 9, which is the constant.
• They add up to -10, which matches the linear coefficient.

Thus, we can factor the expression as \((n-1)(n-9)\). This factorization is instrumental as it allows us to find common factors with the denominator and simplify the expression by canceling out these factors.

Simplifying Expressions

Simplifying expressions involves reducing the expression to its simplest possible form. In the case of rational expressions, this often means canceling out any common factors. In the exercise at hand, after factoring the quadratic \( n^{2}-10n+9 \) as \((n-1)(n-9)\), we rewrite the initial expression as:\[ \frac{(n-1)(n-9)}{n-9} \times \frac{1}{n-1} \]By examining the expression, we can identify and cancel the common factors from the numerator and the denominator.

• The factor \((n-9)\) appears both in the numerator and the denominator and can be canceled.
• The factor \((n-1)\) also appears in both the numerator and the reciprocal denominator and can be canceled.

After canceling all common factors, the expression simplifies to 1. This entire process requires a careful approach to ensure that every possible simplification opportunity is taken advantage of