Problem 31 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{m^{2}+3 m+2}{m^{2}+5 m+4} \div \frac{m^{2}+5 m+6}{m^{2}+10 m+24}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{m+6}{m+3} \).

Step 1: Understand the Problem

Our task is to simplify the quotient \( \frac{m^{2}+3m+2}{m^{2}+5m+4} \div \frac{m^{2}+5m+6}{m^{2}+10m+24} \). To do this, we need to divide one fraction by another, which involves multiplying by the reciprocal of the second fraction.

Step 2: Change Division to Multiplication

Rewrite the expression to change the division into multiplication, which involves flipping the second fraction into its reciprocal:\[\frac{m^{2}+3m+2}{m^{2}+5m+4} \times \frac{m^{2}+10m+24}{m^{2}+5m+6}\]

Step 3: Factor Each Quadratic Expression

Factor each of the quadratic expressions:- \( m^2+3m+2 = (m+1)(m+2) \)- \( m^2+5m+4 = (m+1)(m+4) \)- \( m^2+5m+6 = (m+2)(m+3) \)- \( m^2+10m+24 = (m+4)(m+6) \).Rewrite the expression using these factored forms:

Step 4: Simplify the Expression

Substitute the factored forms into the expression:\[\frac{(m+1)(m+2)}{(m+1)(m+4)} \times \frac{(m+4)(m+6)}{(m+2)(m+3)}\]Cancel out the common factors in the numerator and denominator. \((m+1)\) and \((m+4)\) cancel from both parts of their respective fraction to get:\[\frac{(m+6)}{(m+3)}\].

Step 5: Write the Simplified Result

The simplified form of the original expression is \( \frac{m+6}{m+3} \).

Key Concepts

Factoring Quadratics

When dealing with algebraic fractions, factoring quadratics plays a vital role in simplifying expressions. Quadratic expressions are polynomial equations of the form \( ax^2 + bx + c \). To factor these, we look for two binomials that multiply to give us the original quadratic. This requires finding two numbers that multiply to \( ac \) and add up to \( b \). Let's use the expression \( m^2 + 3m + 2 \) as an example. We need two numbers that multiply to \( 1 \times 2 = 2 \) and add up to 3. The numbers 1 and 2 satisfy these requirements, so we can factor the quadratic as \( (m + 1)(m + 2) \). The same approach applies to other quadratics in the problem:

• \( m^2 + 5m + 4 \) factors to \( (m + 1)(m + 4) \).
• \( m^2 + 5m + 6 \) factors to \( (m + 2)(m + 3) \).
• \( m^2 + 10m + 24 \) factors to \( (m + 4)(m + 6) \).

Factoring helps us transform complex expressions into simpler forms that are easier to manipulate and solve.

Simplifying Algebraic Expressions

Once we've factored the quadratic expressions, the next step is simplifying the algebraic expression. Simplification involves reducing the expression to its simplest form by canceling out common factors.For the expression \[\frac{(m+1)(m+2)}{(m+1)(m+4)} \times \frac{(m+4)(m+6)}{(m+2)(m+3)}\],we can start canceling common factors between numerators and denominators. Here,

• \((m+1)\) appears in both the numerator and the denominator.
• Similarly, \((m+4)\) also appears in both.

Cancel these terms to significantly simplify the expression, reducing it to \[\frac{(m+6)}{(m+3)}\].This process makes the expression more manageable and reveals its true behavior more clearly. Always ensure that you're only canceling terms that appear identically in both parts, and confirm that no zeros are introduced into any denominators.

Division of Fractions

Dividing fractions involves a special operation. Instead of dividing directly, we multiply by the reciprocal of the second fraction. This changes the division into a multiplication problem, simplifying the process.For example, given two fractions,\[\frac{m^2+3m+2}{m^2+5m+4} \div \frac{m^2+5m+6}{m^2+10m+24},\]transform it by flipping the second fraction:\[\frac{m^2+3m+2}{m^2+5m+4} \times \frac{m^2+10m+24}{m^2+5m+6}.\]This method is grounded in the property of reciprocals where \( a \div b = a \times \frac{1}{b} \). The advantage is that multiplication is often easier to handle, especially when we factor and simplify the expressions, enabling us to effectively manage and solve complex fractional algebraic equations.