Question: Simplify each rational expression.
1. $\frac{5y^2 + 11y + 2}{5y^2 + y}$
2. $\frac{20 - 5x}{3x^2 - 48}$
3. $\frac{8k - 10k - 3}{6k^2 - 13k + 6}$
PRACTICE
Simplify each rational expression.
1. $\frac{-4n^2 + 4n - v^2}{2u^2 + uv - v^2}$
2. $\frac{x - 7}{(6x - 4)(x - 7)}$
3. $\frac{x(x^3 + 7x^2) + 2x^2(x + 9)}{x^4 - 2(x^3 + 7x^2) - x^2}$
4. $\frac{-108p^2 - 153p - 54}{96p^2 + 104p + 24}$
1. Simplify $\frac{-4n^2 + 4n - v^2}{2u^2 + uv - v^2}$.
Step 1: State the problem.
We want to simplify the rational expression $\frac{-4n^2 + 4n - v^2}{2u^2 + uv - v^2}$.
Step 2: Factor numerator and denominator if possible.
Numerator: $-4n^2 + 4n - v^2$ cannot be factored further with respect to $n$ and $v$ as variables are different.
Denominator: $2u^2 + uv - v^2$ is a quadratic in $u$.
Step 3: Factor denominator using quadratic formula or factoring by grouping.
Try to factor $2u^2 + uv - v^2$:
Look for factors of $2 \times (-v^2) = -2v^2$ that sum to $u$ coefficient $1v$.
Factors: $2v$ and $-v$ sum to $v$.
Rewrite:
$$2u^2 + 2uv - uv - v^2 = (2u^2 + 2uv) - (uv + v^2) = 2u(u + v) - v(u + v) = (2u - v)(u + v)$$
Step 4: Write the expression with factored denominator:
$$\frac{-4n^2 + 4n - v^2}{(2u - v)(u + v)}$$
Step 5: Since numerator and denominator share no common factors, the expression is simplified as is.
Final answer:
$$\boxed{\frac{-4n^2 + 4n - v^2}{(2u - v)(u + v)}}$$