Problem 124 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{13 x+39}{4 x^{2}+24 x+36} \div\left(\frac{7}{3 x+9}-\frac{5}{4 x+12}\right)$$

Step-by-Step Solution

Answer

The simplified result is 3.

Step 1: Simplify the First Fraction

The first fraction is \( \frac{13x + 39}{4x^2 + 24x + 36} \). We can factor both the numerator and the denominator. The numerator can be factored as \( 13(x + 3) \). The denominator is a perfect square trinomial: \( (2x + 6)^2 \). Therefore, the simplified form of the first fraction is \( \frac{13(x + 3)}{(2x + 6)^2} \).

Step 2: Simplify Inside the Parenthesis

We have the expression \( \frac{7}{3x + 9} - \frac{5}{4x + 12} \). Begin by factoring \( 3x + 9 \) to \( 3(x + 3) \) and \( 4x + 12 \) to \( 4(x + 3) \). The common denominator for these fractions is \( 12(x + 3) \). Rewrite each fraction with this denominator: \(\frac{7}{3(x + 3)} = \frac{28}{12(x + 3)}\) and \(\frac{5}{4(x + 3)} = \frac{15}{12(x + 3)}\). Subtract these fractions to get \( \frac{28 - 15}{12(x + 3)} = \frac{13}{12(x + 3)} \).

Step 3: Divide the Simplified Fractions

We are left with dividing \( \frac{13(x + 3)}{(2x + 6)^2} \) by \( \frac{13}{12(x + 3)} \). Division by a fraction is the same as multiplication by its reciprocal. Thus, multiply \( \frac{13(x + 3)}{(2x + 6)^2} \) by \( \frac{12(x + 3)}{13} \). This simplifies to:\[\frac{13(x + 3)}{(2x + 6)^2} \times \frac{12(x + 3)}{13} = \frac{12(x + 3)^2}{(2x + 6)^2}\]

Step 4: Simplify the Final Expression

The expression \( \frac{12(x + 3)^2}{(2x + 6)^2} \) can be further simplified because \( 2x + 6 = 2(x + 3) \). Thus, \((2x + 6)^2 = 4(x + 3)^2 \). Simplifying gives:\[\frac{12(x + 3)^2}{4(x + 3)^2} = \frac{12}{4} = 3\] The final simplified value is 3.

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. This involves factoring both the numerator and the denominator and then canceling out common factors. Let's consider the fraction \( \frac{13x + 39}{4x^2 + 24x + 36} \).

• First, factor the numerator, which is \( 13(x + 3) \).
• Next, look at the denominator, which is a trinomial. In this case, it is a perfect square trinomial, \( (2x + 6)^2 \).

The simplified fraction becomes \( \frac{13(x + 3)}{(2x + 6)^2} \)._br> Simplifying fractions like this reduces complexity in expressions, making them easier to work with in subsequent operations.

Factoring Trinomials

Factoring trinomials is a key skill in algebra that involves breaking down a polynomial into products of simpler polynomials. In our example, the expression \( 4x^2 + 24x + 36 \) is recognized as a perfect square trinomial.

• This means it can be factored as \( (2x + 6)^2 \), which is essentially \( (2x + 6) \times (2x + 6) \).
• This factoring is beneficial because it reveals the repeated linear factor \( (x + 3) \) after dividing the initial problem.

Understanding how to factor trinomials helps in simplifying complex expressions and solving polynomial equations more efficiently.

Common Denominator

Finding a common denominator is crucial when dealing with operations involving fractions. When subtracting fractions like \( \frac{7}{3x + 9} \) and \( \frac{5}{4x + 12} \), a common denominator allows us to combine them into a single fraction.

• Factor each denominator: \( 3x + 9 = 3(x + 3) \) and \( 4x + 12 = 4(x + 3) \).
• The common denominator is \( 12(x + 3) \).

Using this common denominator, we rewrite and subtract the fractions: \( \frac{28}{12(x + 3)} - \frac{15}{12(x + 3)} = \frac{13}{12(x + 3)}. \)
Finding a common denominator is a powerful technique that simplifies the arithmetic of fractions by aligning them on a common scale.

Polynomial Division

Polynomial division is a process similar to dividing numbers and is used to divide one polynomial by another. In the exercise, we simplify by dividing \( \frac{13(x + 3)}{ (2x + 6)^2} \) by another fraction, \( \frac{13}{12(x + 3)} \).

• To manage this division, multiply by the reciprocal: \( \frac{13(x + 3)}{(2x + 6)^2} \times \frac{12(x + 3)}{13} \).
• This step involves multiplying the fractions instead of directly dividing, simplifying calculations.

This reduces to \( \frac{12(x + 3)^2}{4(x + 3)^2} \), which eventually simplifies to the result \( 3 \).
Understanding polynomial division serves as a strong foundation for working with equations involving polynomial expressions, especially in simplifying and solving algebraic fractions.