Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, add and subtract the rational expressions, and then simplify. \(\frac{x-1}{x+1}-\frac{2 x+3}{2 x+1}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the problem

We are given two rational expressions that need to be subtracted: \( \frac{x-1}{x+1} - \frac{2x+3}{2x+1} \). Our goal is to combine these expressions into a single expression and simplify it.

Step 2: Find a common denominator

Since the denominators are \(x+1\) and \(2x+1\), the common denominator will be \((x+1)(2x+1)\). We'll convert each rational expression to have this common denominator.

Step 3: Rewrite fractions with the common denominator

Rewrite each fraction with the common denominator:\[ \frac{x-1}{x+1} = \frac{(x-1)(2x+1)}{(x+1)(2x+1)} \]\[ \frac{2x+3}{2x+1} = \frac{(2x+3)(x+1)}{(2x+1)(x+1)} \]

Step 4: Expand the numerators

Expand the numerators in each expression:For \((x-1)(2x+1)\), we have:\[ x(2x) + x(1) - 1(2x) - 1(1) = 2x^2 + x - 2x - 1 = 2x^2 - x - 1 \]For \((2x+3)(x+1)\), we have:\[ 2x(x) + 2x(1) + 3(x) + 3(1) = 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3 \]

Step 5: Subtract the numerators

Now subtract the second numerator from the first:\((2x^2 - x - 1) - (2x^2 + 5x + 3)\)This results in:\[ 2x^2 - x - 1 - 2x^2 - 5x - 3 = -6x - 4 \]

Step 6: Write the resulting expression

The resulting expression is:\[ \frac{-6x - 4}{(x+1)(2x+1)} \]

Step 7: Simplify the expression

Factor out the greatest common factor from the numerator:\[ -6x - 4 = -2(3x + 2) \]Then, the simplified expression becomes:\[ \frac{-2(3x + 2)}{(x+1)(2x+1)} \]

Step 8: Final verification

Verify that no further simplification is possible. Since there are no common factors between the numerator \(-2(3x + 2)\) and the denominator \((x+1)(2x+1)\), the expression is fully simplified.

Key Concepts

Common Denominator

When adding or subtracting rational expressions, finding a common denominator is crucial. A common denominator allows us to combine the expressions into a single fraction. Think of the common denominator as a common ground that lets different fractions communicate with each other.

To find the common denominator between two rational expressions, we typically look for the least common multiple (LCM) of the individual denominators. For example, given the denominators \(x+1\) and \(2x+1\), our job is to multiply these together. This gives us a common denominator of \((x+1)(2x+1)\).

• Determine the denominators: Identify the denominators of each expression.
• Multiply: If the denominators have no common factors, multiply them together directly.
• Form the new fraction: Rewrite each fraction using this new common denominator by adjusting the numerators accordingly.

Simplifying Fractions

Simplifying fractions involves making a complex fraction easier by reducing it to its simplest form. In the context of rational expressions, simplification helps in breaking down polynomials or compound fractions. It's an essential step to ensure your answers are presented neatly.

Typically, simplifying involves factoring the numerator and the denominator as much as possible. This helps in identifying any common factors that can be cancelled out.

• Factor the polynomial: Break down both the numerator and denominator into their factors.
• Cancel common factors: If both the numerator and the denominator share common factors, cancel them.
• Final expression: Write the fraction with the remaining factors to get the simplified version.

After subtraction in our example, the numerator became \(-6x - 4\). By factoring, we see that \(-2\) is a common factor, resulting in \(-2(3x + 2)\), which gives the simplified rational expression \(\frac{-2(3x + 2)}{(x+1)(2x+1)}\).

Polynomial Expansion

The concept of polynomial expansion is key when dealing with subtracting rational expressions. It involves multiplying expressions within parentheses so that you express them as a sum of terms. It helps in making the numerators in a common denominator easily comparable.

In our case, two rectangular binomials \((x-1)\) and \((2x+1)\), and then \((2x+3)\) and \((x+1)\), were expanded in terms of \(x\) to achieve comparable numerators:

• Distribute each term: Multiply every term in the first polynomial by every term in the second polynomial.
• Combine like terms: After expanding, combine any like terms to simplify the expression.
• Write the expanded form: The polynomial expansion should clearly show all the terms without any parentheses.

For example, expanding \((x-1)(2x+1)\) led to \(2x^2 - x - 1\), and expanding \((2x+3)(x+1)\) resulted in \(2x^2 + 5x + 3\). These expanded forms allow for straightforward subtraction.