Problem 106 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{x+3}{2 x^{2}-5 x+2}-\frac{3 x-1}{x^{2}-x-2}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{-5x^2 + 9x + 2}{(2x - 1)(x - 2)(x + 1)}\), as further factoring is not possible.

Step 1: Simplify Denominators

Start by factoring the denominators. For the first fraction, the expression is \(2x^2 - 5x + 2\). The factors of this are \((2x - 1)(x - 2)\). For the second fraction, the expression is \(x^2 - x - 2\). The factors of this are \((x - 2)(x + 1)\).

Step 2: Find a Common Denominator

To subtract the fractions, find a common denominator by multiplying the denominators' factors. The common denominator of \((2x - 1)(x - 2)\) and \((x - 2)(x + 1)\) is \((2x - 1)(x - 2)(x + 1)\).

Step 3: Convert to Common Denominator

Adjust the numerators to express both fractions over the common denominator. Multiply the numerator \(x + 3\) by the missing factor \((x + 1)\) from \((2x - 1)(x - 2)(x + 1)\). Then, multiply the numerator \(3x - 1\) by the missing factor \((2x - 1)\).

Step 4: Expand the New Numerators

For the first fraction: \((x + 3)(x + 1) = x^2 + 4x + 3\). For the second fraction: \((3x - 1)(2x - 1) = 6x^2 - 5x + 1\).

Step 5: Perform the Subtraction

Subtract the second adjusted numerator from the first: \((x^2 + 4x + 3) - (6x^2 - 5x + 1)\). Simplify by distributing and combining like terms to get \(-5x^2 + 9x + 2\). This results in the fraction \(\frac{-5x^2 + 9x + 2}{(2x - 1)(x - 2)(x + 1)}\).

Step 6: Simplify If Possible

Check if the numerator \(-5x^2 + 9x + 2\) can be factored further for simplification. If no common factors exist, then the fraction is already in its simplest form.

Key Concepts

Factoring Polynomials

Factoring polynomials is a key skill in working with rational expressions, because it allows us to break down complex expressions into simpler, manageable parts. By identifying factors, we can simplify expressions, find common denominators, and eventually solve the problem at hand.

To factor a polynomial, examine the expression to find pairs of terms that can be multiplied to recreate the original polynomial. The given problem involves factoring the quadratics in the denominators:

• For the expression \(2x^2 - 5x + 2\), factor it into \((2x - 1)(x - 2)\).
• For the expression \(x^2 - x - 2\), the factors are \((x - 2)(x + 1)\).

These factored forms are essential as they not only help us find the common denominator but also make future operations much easier. Noticing the common terms, like \((x - 2)\) here, is crucial during this process.

Common Denominator

Finding a common denominator is vital when adding or subtracting rational expressions. A common denominator allows us to combine fractions by equating their bases; this makes arithmetic involving fractions straightforward.

In the given problem, the denominators after factoring are \((2x - 1)(x - 2)\) and \((x - 2)(x + 1)\). To find a common denominator:

• Identify all unique factors. Here, they are \((2x - 1)\), \((x - 2)\), and \((x + 1)\).
• Combine them to form the common denominator: \((2x - 1)(x - 2)(x + 1)\).

This common denominator allows you to rewrite each fraction so they can be combined. The choice of a common denominator reflects all component factors of the original denominators, ensuring each fraction can be rewritten equivalently over the same base. This is the gateway for performing subtraction or addition.

Subtracting Fractions

Once you have a common denominator, subtracting fractions becomes straightforward. The key is to adjust the numerators accordingly, ensuring each fraction reflects the equivalent expression over the new denominator.

In our problem, adjust each numerator using the missing factors:

• The numerator \(x + 3\) needs to be multiplied by \((x + 1)\).
• The numerator \(3x - 1\) needs to be multiplied by \((2x - 1)\).

This adjustment gives us equivalent expressions over the same common denominator.Now, you can subtract one numerator from the other:

• Expand \((x + 3)(x + 1)\) to get \(x^2 + 4x + 3\).
• Expand \((3x - 1)(2x - 1)\) to get \(6x^2 - 5x + 1\).
• Subtract these: \((x^2 + 4x + 3) - (6x^2 - 5x + 1)\).

By carefully doing subtraction, we derive the simplified numerator \(-5x^2 + 9x + 2\), which is pivotal for further simplification.

Simplifying Algebraic Expressions

Simplification is the ultimate goal in working with rational expressions. Simplifying involves reducing expressions to their most concise form, making them easier to interpret and use.

Once you perform the subtraction in the numerators, you have \(-5x^2 + 9x + 2\) over the common denominator \((2x - 1)(x - 2)(x + 1)\). At this point, check if the numerator can be factored further:

• Attempt to find any common factors by looking at the coefficients and constant terms.
• If no factors are found, as with \(-5x^2 + 9x + 2\), the expression is in its simplest form.

Simplifying is about efficiency in solutions, reducing the complexity wherever possible. Through careful checking, you ensure you're working with the most direct expression possible, ready for any further mathematical or algebraic application needed.