Problem 87 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{2 z^{2}-3 z+6}{z^{2}-1}-\frac{z^{2}-5 z+9}{z^{2}-1} $$

Step-by-Step Solution

Answer

\( \frac{z + 3}{z + 1} \)

Step 1: Identify the Common Denominator

The given fractions have a common denominator of \( z^2 - 1 \). Recognize that both fractions can be combined using the same denominator.

Step 2: Combine the Numerators

Since the denominators are the same, the numerators can be combined: \[ \frac{2z^2 - 3z + 6 - (z^2 - 5z + 9)}{z^2 - 1} \].

Step 3: Distribute the Negative Sign

Distribute the negative sign in the numerator: \[ 2z^2 - 3z + 6 - z^2 + 5z - 9. \]

Step 4: Simplify the Numerator

Combine like terms in the numerator: \[ (2z^2 - z^2) + (-3z + 5z) + (6 - 9) = z^2 + 2z - 3. \]

Step 5: Write the Simplified Expression

The simplified form of the fraction is: \[ \frac{z^2 + 2z - 3}{z^2 - 1}. \]

Step 6: Factor the Numerator and Denominator

Factor the numerator and the denominator if possible. \[ z^2 + 2z - 3 = (z + 3)(z - 1) \] and \[ z^2 - 1 = (z + 1)(z - 1). \]

Step 7: Cancel Common Factors

Cancel the common factor \( z - 1 \) in the numerator and denominator. The simplified expression is: \[ \frac{z + 3}{z + 1}. \]

Key Concepts

Common Denominator

When dealing with rational expressions, finding a common denominator is crucial. This allows us to combine fractions more easily. In this exercise, both fractions already share the same denominator: \( z^2 - 1 \). The common denominator tells us the expression beneath the division line is the same for both fractions. This makes our job easier because we don't need to perform any further steps to match the denominators before we can combine the fractions.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable raised to the same power. This is often seen in the numerators of rational expressions. In our exercise, once we recognize that the fractions have a common denominator, we combine the numerators: \( 2z^2 - 3z + 6 \) and \( -(z^2 - 5z + 9) \). Remember to distribute the negative sign to each term inside the parentheses before combining:

• \( 2z^2 - z^2 \)
• \( -3z + 5z \)
• \( 6 - 9 \)

Adding these like terms simplifies to \( z^2 + 2z - 3 \).

Simplifying Fractions

Once combined, we must simplify the fractions. Simplification involves reducing the expression to its simplest form. For our problem, we have \( \frac{z^2 + 2z - 3}{z^2 - 1} \). Here, 'simplifying' means to check if we can factor both the numerator and denominator. Factoring often reveals common factors that we can cancel out to further reduce the expression.

Factoring Polynomials

Factoring polynomials is a method to express a polynomial as a product of smaller polynomials. This helps in simplifying and canceling terms in rational expressions. In our example, we factor the numerator and denominator:

• For the numerator \( z^2 + 2z - 3 \), we get \( (z + 3)(z - 1) \).
• For the denominator \( z^2 - 1 \), we get \( (z + 1)(z - 1) \).

Factoring makes it easier to see common factors that can be canceled out.

Canceling Common Factors

Once we factor both the numerator and the denominator, we look for common factors that appear in both. In our scenario, the common factor is \( z - 1 \). Canceling common factors means removing the same term from the numerator and denominator to simplify the fraction further. This leaves us with:
\[ \frac{(z + 3)(z - 1)}{(z + 1)(z - 1)} = \frac{z + 3}{z + 1} \] Always ensure that canceling is only done with factors, not terms. This is an important distinction in avoiding mistakes.