Questions: Simplify the expression: (v^2 + 6v + 5) / (6v^2 + 12v + 6)

Transcript text: Rational Expressions Simplifying a ratio of polynomials: Problem type 1 Simplify. \[ \frac{v^{2}+6 v+5}{6 v^{2}+12 v+6} \]

Solution

Solution Steps

To simplify the given rational expression, we need to factor both the numerator and the denominator. Once factored, we can cancel out any common factors between the numerator and the denominator.

Step 1: Factor the Numerator

The numerator \( v^2 + 6v + 5 \) can be factored as: \[ v^2 + 6v + 5 = (v + 1)(v + 5) \]

Step 2: Factor the Denominator

The denominator \( 6v^2 + 12v + 6 \) can be factored as: \[ 6v^2 + 12v + 6 = 6(v + 1)^2 \]

Step 3: Simplify the Rational Expression

Now, we can write the original rational expression as: \[ \frac{v^2 + 6v + 5}{6v^2 + 12v + 6} = \frac{(v + 1)(v + 5)}{6(v + 1)^2} \] We can cancel the common factor \( (v + 1) \) from the numerator and the denominator: \[ \frac{(v + 5)}{6(v + 1)} \]