Problem 86 - Algebra for College Students

Question

$$ \frac{w^{2}-3 w+6}{w-5}+\frac{9-w^{2}}{w-5} $$$$ \frac{2 z^{2}-3 z+6}{z^{2}-1}-\frac{z^{2}-5 z+9}{z^{2}-1} $$

Step-by-Step Solution

Answer

-3, \frac{z + 3}{z + 1}

Step 1 - Combine Like Denominators (First Expression)

Notice that both fractions in the first expression have the same denominator, \(w-5\). Combine the numerators: \ \ \[ \frac{w^2 - 3w + 6 + 9 - w^2}{w-5} \]

Step 2 - Simplify the Numerator (First Expression)

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

Step 3 - Simplify Further (First Expression)

Simplify the numerator further: \ \ \[ \frac{-3(w - 5)}{w-5} = -3\]

Step 4 - Combine Like Denominators (Second Expression)

For the second expression, notice both fractions have the same denominator, \(z^2 - 1 = (z - 1)(z + 1)\). Combine the numerators: \ \ \[ \frac{2z^2 - 3z + 6 - (z^2 - 5z + 9)}{z^2 - 1} \]

Step 5 - Simplify the Numerator (Second Expression)

Distribute the minus sign and combine the terms in the numerator: \ \ \[ \frac{2z^2 - 3z + 6 - z^2 + 5z - 9}{(z - 1)(z + 1)} = \frac{z^2 + 2z - 3}{(z - 1)(z + 1)} \]

Step 6 - Factor and Simplify (Second Expression)

Factor the numerator: \ \ \[ \frac{(z + 3)(z - 1)}{(z - 1)(z + 1)} \]. Cancel the \(z - 1\) terms: \ \ \[ \frac{z + 3}{z + 1} \]

Key Concepts

algebraic fractions

Algebraic fractions are fractions where both the numerator and the denominator are polynomials. Just like regular fractions, you can add, subtract, multiply, and divide them. However, working with algebraic fractions often involves additional steps like factoring and finding a common denominator.
For example, in the original exercise with the first expression:
\[ \frac{w^{2}-3w+6}{w-5}+\frac{9-w^{2}}{w-5} \]
Both fractions had the same denominator, making it easier to combine them. Always aim to have a common denominator when adding or subtracting algebraic fractions.

simplifying rational expressions

Simplifying rational expressions involves reducing them to their simplest form by canceling common factors from the numerator and the denominator.
In the first expression's final simplification step:
\[ \frac{-3(w - 5)}{w-5} = -3 \]
We noticed that the numerator \( -3(w - 5) \) and the denominator \( w-5 \) share a common factor. Canceling this factor gives \ -3 \ as the simplified result.
Remember to factor both the numerator and the denominator completely. This way, you can easily spot and cancel any common factors.

factoring

Factoring is breaking down a polynomial into simpler polynomials, or 'factors', that when multiplied together give the original polynomial.
In the solution's second expression:
\[ \frac{(z + 3)(z - 1)}{(z - 1)(z + 1)} \]
we factored the numerator and denominator. By recognizing that \( z^2 - 1 \) can be factored into \( (z-1)(z+1) \), we were able to simplify the expression.
Always look for common factors or special polynomials (like the difference of squares, trinomials, etc.) when factoring. This helps in simplifying the rational expression significantly.