Question

directions: simplify each of the following rational expressions.

1. \\(\frac{6y - 18}{5y^2 - 13y - 6}\\)
2. \\(\frac{x^2 - 16}{x^2 - 6x + 8}\\)

Explanation:

Problem 1: Simplify \(\boldsymbol{\frac{6y - 18}{5y^2 - 13y - 6}}\)

Step 1: Factor numerator and denominator

• Numerator: Factor out 6 from \(6y - 18\).

\(6y - 18 = 6(y - 3)\)

• Denominator: Factor \(5y^2 - 13y - 6\) (using AC method: \(5 \times (-6) = -30\), find two numbers that multiply to -30 and add to -13: -15 and 2).

\(5y^2 - 15y + 2y - 6 = 5y(y - 3) + 2(y - 3) = (5y + 2)(y - 3)\)

Step 2: Cancel common factors

The simplified expression is:
\(\frac{6(y - 3)}{(5y + 2)(y - 3)} = \frac{6}{5y + 2}\) (since \(y
eq 3\) to avoid division by zero)

Problem 2: Simplify \(\boldsymbol{\frac{x^2 - 16}{x^2 - 6x + 8}}\)

Step 1: Factor numerator and denominator

• Numerator: Use difference of squares (\(a^2 - b^2 = (a - b)(a + b)\)):

\(x^2 - 16 = (x - 4)(x + 4)\)

• Denominator: Factor \(x^2 - 6x + 8\) (find two numbers that multiply to 8 and add to -6: -2 and -4).

\(x^2 - 6x + 8 = (x - 2)(x - 4)\)

Step 2: Cancel common factors

The simplified expression is:
\(\frac{(x - 4)(x + 4)}{(x - 2)(x - 4)} = \frac{x + 4}{x - 2}\) (since \(x
eq 4\) to avoid division by zero)

Answer:

s:

1. \(\frac{6}{5y + 2}\)
2. \(\frac{x + 4}{x - 2}\)