Simplify the following \( \frac{\text { expressions }}{\text { a) } \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6}} \) b) \( \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \)
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To simplify the given expressions, we'll tackle them one at a time. For part (a): Starting with the expression \( \frac{x^{2}+5x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5x+6} \), we can rewrite the division as multiplication by the reciprocal: \[ \frac{x^{2}+5x+6}{x^{2}-9} \times \frac{x^{2}-5x+6}{x^{2}-4} \] Next, let's factor these polynomials. 1. \( x^2 + 5x + 6 = (x+2)(x+3) \) 2. \( x^2 - 9 = (x-3)(x+3) \) 3. \( x^2 - 4 = (x-2)(x+2) \) 4. \( x^2 - 5x + 6 = (x-2)(x-3) \) Rewriting the expression with these factors gives: \[ \frac{(x+2)(x+3)}{(x-3)(x+3)} \times \frac{(x-2)(x-3)}{(x-2)(x+2)} \] Now, we can cancel out the common factors: - The \( (x+2) \) cancels. - The \( (x-2) \) cancels. - The \( (x+3) \) cancels. This leaves us with: \[ \frac{(x-3)}{(x-3)} = 1 \quad \text{(as long as } x \neq 3 \text{)} \] For part (b): We start with the expression \[ \frac{x}{x-2} - \frac{1}{x+2} + \frac{8}{4-x^{2}} \] Let's factor \( 4 - x^2 \) first, which is a difference of squares: \[ 4 - x^2 = (2-x)(2+x) \] Now rewriting the entire expression, we bring everything under a common denominator which is \( (x-2)(x+2)(2-x)(2+x) \). Transform \( \frac{8}{4-x^2} \) as: \[ \frac{8}{(2-x)(2+x)} = \frac{-8}{(x-2)(2+x)} \quad \text{(since } 2-x = -(x-2) \text{)} \] Now, applying the common denominator: 1. For \( \frac{x}{x-2} \), multiply by \( (x+2)(2-x) \): \[ \frac{x((x+2)(2-x))}{(x-2)(x+2)(2-x)(2+x)} \] 2. For \( -\frac{1}{x+2} \), multiply by \( (x-2)(2-x) \): \[ -\frac{(x-2)(2-x)}{(x-2)(x+2)(2-x)(2+x)} \] 3. For \( -\frac{8}{4-x^2} \) (now written correctly): \[ -\frac{8(x-2)}{(x-2)(x+2)(2-x)(2+x)} \] Combining gives: \[ \frac{x((x+2)(2-x)) - (x-2)(2-x) - 8(x-2)}{(x-2)(x+2)(2-x)(2+x)} \] Now, simplifying the numerator will lead to the final expression. Be careful with errors when arranging terms, and once simplified, ensure to factor or cancel where possible.
