Problem 85 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

-3

Step 1: Combine the Fractions

Both fractions have the same denominator, so they can be combined into one single fraction: \[ \frac{w^{2}-3w+6 + (9-w^{2})}{w-5} \]

Step 2: Simplify the Numerator

Combine like terms in the numerator:\[ (w^{2} - w^{2}) + (-3w) + (6 + 9) \]This simplifies to:\[ \frac{-3w + 15}{w-5} \]

Step 3: Factor and Simplify

Factor out common terms from the numerator:\[ \frac{-3(w - 5)}{w-5} \]Since the numerator and the denominator have a common factor, cancel it out:\[ -3 \]

Key Concepts

Combining Fractions

When dealing with fraction operations in algebra, combining fractions often simplifies your work. In this problem, the fractions have the same denominator, which simplifies things a lot. You can combine them by adding their numerators together while keeping the denominator the same. So, for the expression \[ \frac{w^{2}-3w+6}{w-5}+\frac{9-w^{2}}{w-5} \], you add the numerators: \[ w^{2} - 3w + 6 + 9 - w^{2} \]. This combined expression helps reduce the number of terms you have to work with in later steps.

Simplifying Numerators

Simplifying the numerator is a crucial step in making the fraction easier to work with. After combining the numerators in the previous step, you will get: \[ \frac{(w^{2} - w^{2}) + (-3w) + (6 + 9)}{w-5} \]. Here, you can see some like terms that can be simplified. The terms \[ w^{2} \] and \[ -w^{2} \] cancel each other out, leaving you with: \[ -3w + 15 \]. Simplifying like this helps in making the fraction less complicated.

Factoring and Canceling Terms

The last step is to factor and cancel common terms. The simplified numerator from the previous step is \[ -3w + 15 \]. You can factor out a \[ -3 \] from this expression, which gives you: \[ -3(w - 5) \]. Now, you have \[ \frac{-3(w-5)}{w-5} \]. Since the numerator and the denominator both contain a \[ w-5 \], you can cancel this common term. The expression simplifies to: \[ -3 \]. By simplifying early and carefully at each step, you make your work much easier.