Problem 56 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{r^{2}+12 r+11}{r} \div \frac{r+11}{r}\)

Step-by-Step Solution

Answer

The simplified result is \(r+1\).

Step 1: Identify Division of Fractions

The expression involves division of two fractions. When dividing fractions, multiply by the reciprocal of the second fraction. Thus, \( \frac{r^{2}+12r+11}{r} \div \frac{r+11}{r} \) becomes \( \frac{r^{2}+12r+11}{r} \times \frac{r}{r+11} \).

Step 2: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together. The result is: \( \frac{(r^{2}+12r+11) \times r}{r \times (r+11)} \).

Step 3: Simplify the Expression

First, notice that \(r\) in the numerator and denominator cancels out. This reduces the expression to:\( \frac{r^{2}+12r+11}{r+11} \).

Step 4: Factor the Numerator

Factor \(r^{2}+12r+11\) into \((r+11)(r+1)\). This gives us:\( \frac{(r+11)(r+1)}{r+11} \).

Step 5: Simplify by Canceling Common Factors

Cancel the common factor \(r+11\) in the numerator and the denominator. This reduces the expression to: \(r+1\).

Step 6: State the Final Result

After simplifying, the result of the division is \(r+1\).

Key Concepts

Understanding Fractions

In mathematics, a fraction represents a part of a whole or a ratio between numbers. Fractions are composed of two parts: the numerator, located above the line, and the denominator, found below it. When dividing fractions, the operation involves flipping the second fraction, known as the reciprocal, and then proceeding to multiply. Fractions are a fundamental concept in algebra and allow us to express quantities that aren't whole numbers. Working with fractions involves skills like finding common denominators, simplifying by canceling common terms, and understanding how fractions can represent different forms of the same value.

• Numerator: the top part of a fraction, showing how many parts are taken.
• Denominator: the bottom part of a fraction, indicating the total number of equal parts.

Fractions can sometimes be intimidating, but they provide essential groundwork for algebraic methods and division.

Exploring Reciprocals

A reciprocal is essentially what you get when you "flip" a fraction. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This concept is key when dividing fractions. Instead of dividing directly, you multiply by the reciprocal of the second fraction. This is because multiplying by a reciprocal gives a result of one, effectively simplifying calculations.

• Flipping: Switch the numerator and the denominator to find the reciprocal.
• Application: Use reciprocals to transform division into multiplication.

The concept of reciprocals is useful beyond fractions. It aids in solving equations and understanding relationships between numbers or expressions.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler terms called factors that, when multiplied together, give back the original polynomial. For example, the polynomial \( r^2 + 12r + 11 \) can be factored into \( (r + 11)(r + 1) \). This step lets us simplify complex algebraic expressions.

• Finding Factors: Look for two numbers that multiply to give the constant term, and add up to the coefficient of the middle term.
• Check Your Work: Multiply the factors to ensure they result in the original polynomial.

Factoring is crucial in algebra, particularly for simplifying expressions and solving equations.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its simplest form. This often involves canceling out terms or factors that appear in both the numerator and the denominator. In the given exercise, this happened when the factor \( r+11 \) appeared in both the numerator and denominator, making it possible to cancel out.

• Cancel Common Factors: Search for factors that appear in both parts of a fraction, and eliminate them to simplify.
• Always Verify: Check to make sure terms have been properly canceled and the expression is at its simplest form.

Through simplification, expressions become easier to handle and understand while retaining their original value.