Problem 38 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{4 x}{6 x^{2}+7 x+2}-\frac{2}{2 x+1}-\frac{4}{3 x+2} $$

Step-by-Step Solution

Answer

\( \frac{-10x - 8}{(2x+1)(3x+2)} \)

Step 1: Identify the Common Denominator

The given expression is \[ \frac{4x}{6x^2+7x+2} - \frac{2}{2x+1} - \frac{4}{3x+2} \] First, factor the quadratic expression in the first fraction's denominator into linear terms: \[ 6x^2+7x+2 = (2x+1)(3x+2) \] Hence, the common denominator needed for all fractions is \[ (2x+1)(3x+2) \]

Step 2: Rewrite Fractions with Common Denominator

Rewrite each fraction using the common denominator:1. The first fraction is already using the common denominator: \[ \frac{4x}{(2x+1)(3x+2)} \]2. Rewrite the second fraction using the common denominator: \[ \frac{2}{2x+1} = \frac{2(3x+2)}{(2x+1)(3x+2)} = \frac{6x+4}{(2x+1)(3x+2)} \]3. Rewrite the third fraction using the common denominator: \[ \frac{4}{3x+2} = \frac{4(2x+1)}{(3x+2)(2x+1)} = \frac{8x+4}{(3x+2)(2x+1)} \]

Step 3: Combine the Fractions

Now that all fractions have the same denominator, combine them:\[\frac{4x}{(2x+1)(3x+2)} - \frac{6x+4}{(2x+1)(3x+2)} - \frac{8x+4}{(3x+2)(2x+1)} = \frac{4x - (6x+4) - (8x+4)}{(2x+1)(3x+2)}\]

Step 4: Simplify the Numerator

Simplify the expression in the numerator:\[4x - 6x - 4 - 8x - 4 = -10x - 8\]Thus, the expression becomes:\[\frac{-10x - 8}{(2x+1)(3x+2)}\]

Step 5: Simplify Further if Possible

The expression \(-10x - 8\) can be factored as \(-2(5x + 4)\), but since \(5x + 4\) is not a factor of the denominator, no further simplification is possible. The final simplified expression is:\[\frac{-10x - 8}{(2x+1)(3x+2)}\]

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra. The goal is to express the fraction in its simplest form until no further reduction is possible.
A fraction consists of a numerator at the top and a denominator at the bottom. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

• Identify factors of both numerator and denominator.
• Find the GCD.
• Divide both by the GCD to simplify.

Understanding how to simplify fractions is crucial, especially when working with complex algebraic expressions.
Simplifying can help you see relationships in numbers and make calculations easier.

Common Denominator

Finding a common denominator allows you to combine and compare fractions easily. It is especially important when you deal with fractions that have different denominators.
To find a common denominator:

• Factor each denominator to its prime factors.
• Identify the least common multiple (LCM) of these factors.

In the given exercise, the denominators were factored and a common denominator was found in \((2x+1)(3x+2)\). This step is necessary in order to add or subtract fractions.
Once a common denominator is found, rewrite each fraction to use it. This alignment allows for direct combination of numerators.

Factoring Polynomials

Factoring polynomials is the method of breaking down a polynomial into simpler polynomials that multiply together to get the original polynomial. It aids in finding common denominators and simplifying expressions.
Consider a quadratic polynomial like \(6x^2 + 7x + 2\). To factor it:

• Look for two numbers that multiply to the constant term \(c\) and add up to the middle coefficient \(b\).
• Rewrite the middle term using these numbers and factor by grouping.

Factoring \(6x^2 + 7x + 2\) yields \((2x+1)(3x+2)\), which was used as a common denominator in the exercise.
Proper factoring is fundamental in algebraic manipulation.

Rational Expressions

Rational expressions are ratios of two polynomials. Manipulating them involves operations such as addition, subtraction, multiplication, or division.
In the exercise, you are subtracting multiple rational expressions. Proper understanding of their properties is key to solving such expressions correctly.
Just like fractions, operations with rational expressions require a common denominator. Simplification might involve factoring or canceling common terms when possible.

• Always start by factoring each piece of the expression.
• Use common denominators for addition or subtraction.
• Simplify whenever possible by canceling common factors.

Through a clear understanding of rational expressions, algebraic problems become manageable and solvable.