Question

box your final answer.

1. simplify completely: $\frac{x - 5}{x + 5} - \frac{3x - 21}{x + 5}$. (3 pts)
2. simplify completely: $\frac{x}{x^2 - 1} + \frac{2}{x^2 - 3x + 2}$. (5 pts)
3. name the lcm for: $x^2 - 2x - 15$ and $x^2 - 5x$. (3 pts)

Explanation:

Problem 1

Step1: Subtract numerators (same denominator)

Since the denominators are the same ($x + 5$), we subtract the numerators: $\frac{(x - 5) - (3x - 21)}{x + 5}$

Step2: Simplify numerator

Expand and combine like terms in the numerator: $x - 5 - 3x + 21 = -2x + 16$

Step3: Factor numerator (if possible)

Factor out -2 from the numerator: $\frac{-2x + 16}{x + 5} = \frac{-2(x - 8)}{x + 5}$ or $\frac{2(8 - x)}{x + 5}$ (both are simplified)

Step1: Factor denominators

Factor $x^2 - 1$ as $(x - 1)(x + 1)$ and $x^2 - 3x + 2$ as $(x - 1)(x - 2)$

Step2: Find common denominator

The common denominator is $(x - 1)(x + 1)(x - 2)$

Step3: Rewrite fractions with common denominator

$\frac{x}{(x - 1)(x + 1)} = \frac{x(x - 2)}{(x - 1)(x + 1)(x - 2)}$ and $\frac{2}{(x - 1)(x - 2)} = \frac{2(x + 1)}{(x - 1)(x + 1)(x - 2)}$

Step4: Add numerators

$x(x - 2) + 2(x + 1) = x^2 - 2x + 2x + 2 = x^2 + 2$

Step5: Write final fraction

$\frac{x^2 + 2}{(x - 1)(x + 1)(x - 2)}$ (or expand denominator: $\frac{x^2 + 2}{x^3 - 2x^2 - x + 2}$)

Step1: Factor first polynomial

Factor $x^2 - 2x - 15$: find two numbers with product -15 and sum -2, which are -5 and 3. So, $x^2 - 2x - 15 = (x - 5)(x + 3)$

Step2: Factor second polynomial

Factor $x^2 - 5x$: factor out x, so $x^2 - 5x = x(x - 5)$

Step3: Find LCM of factors

The factors are $x$, $(x - 5)$, $(x + 3)$. LCM is the product of the highest power of each unique factor: $x(x - 5)(x + 3)$

Answer:

$\boxed{\dfrac{-2x + 16}{x + 5}}$ (or equivalent factored form)

Problem 2