#17-21. perform the indicated operations. simplify your answers completely. you may leave your answers in factored form if you wish.

17. $\frac{4}{x^2 - 2x - 8} + \frac{5}{x^2 + 5x + 6}$

18. $\frac{x - 2}{x^2 + 2x - 8} + \frac{2}{x + 4}$

19. $\frac{x + 3}{x^2 - 1} + \frac{1}{x + 1}$

20. $\frac{2x + 1}{x^2 + x} - \frac{1}{x + 1}$

21. $\frac{x + 4}{x^2 + 5x} + \frac{x + 1}{x^2 + 6x + 5}$

Question

#17-21. perform the indicated operations. simplify your answers completely. you may leave your answers in factored form if you wish.

17. $\frac{4}{x^2 - 2x - 8} + \frac{5}{x^2 + 5x + 6}$

18. $\frac{x - 2}{x^2 + 2x - 8} + \frac{2}{x + 4}$

19. $\frac{x + 3}{x^2 - 1} + \frac{1}{x + 1}$

20. $\frac{2x + 1}{x^2 + x} - \frac{1}{x + 1}$

21. $\frac{x + 4}{x^2 + 5x} + \frac{x + 1}{x^2 + 6x + 5}$

Accepted Answer

{ "quality": "clear", "question_count": 5, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Expressions", "Finding Common Denominators", "Factoring Quadratics" ], "new_concepts": [], "current_concepts": [ "Rational Expressions", "Finding Common Denominators", "Factoring Quadratics" ] } ## Factor denominators and find common denominators $$ \begin{aligned} \text{17. } & \frac{4}{x^2-2x-8} + \frac{5}{x^2+5x+6} = \frac{4}{(x-4)(x+2)} + \frac{5}{(x+2)(x+3)} \ & \text{LCD: } (x-4)(x+2)(x+3) \ \text{18. } & \frac{x-2}{x^2+2x-8} + \frac{2}{x+4} = \frac{x-2}{(x-2)(x+4)} + \frac{2}{x+4} \ & \text{LCD: } (x-2)(x+4) \ \text{19. } & \frac{x+3}{x^2-1} + \frac{1}{x+1} = \frac{x+3}{(x-1)(x+1)} + \frac{1}{x+1} \ & \text{LCD: } (x-1)(x+1) \ \text{20. } & \frac{2x+1}{x^2+x} - \frac{1}{x+1} = \frac{2x+1}{x(x+1)} - \frac{1}{x+1} \ & \text{LCD: } x(x+1) \ \text{21. } & \frac{x+4}{x^2+5x} + \frac{x+1}{x^2+6x+5} = \frac{x+4}{x(x+5)} + \frac{x+1}{(x+1)(x+5)} \ & \text{LCD: } x(x+5)(x+1) \end{aligned} $$ ## Rewrite with common denominators and combine numerators $$ \begin{aligned} \text{17. } & \frac{4(x+3) + 5(x-4)}{(x-4)(x+2)(x+3)} = \frac{4x+12+5x-20}{(x-4)(x+2)(x+3)} = \frac{9x-8}{(x-4)(x+2)(x+3)} \ \text{18. } & \frac{x-2 + 2(x-2)}{(x-2)(x+4)} = \frac{3(x-2)}{(x-2)(x+4)} \ \text{19. } & \frac{x+3 + 1(x-1)}{(x-1)(x+1)} = \frac{2x+2}{(x-1)(x+1)} \ \text{20. } & \frac{2x+1 - 1(x)}{x(x+1)} = \frac{x+1}{x(x+1)} \ \text{21. } & \frac{(x+4)(x+1) + x(x+1)}{x(x+5)(x+1)} = \frac{(x+1)(x+4+x)}{x(x+5)(x+1)} = \frac{(x+1)(2x+4)}{x(x+5)(x+1)} \end{aligned} $$ ## Simplify final expressions $$ \begin{aligned} \text{17. } & \frac{9x-8}{(x-4)(x+2)(x+3)} \ \text{18. } & \frac{3}{x+4} \quad (x eq 2) \ \text{19. } & \frac{2(x+1)}{(x-1)(x+1)} = \frac{2}{x-1} \quad (x eq -1) \ \text{20. } & \frac{1}{x} \quad (x eq -1) \ \text{21. } & \frac{2(x+2)}{x(x+5)} \quad (x eq -1) \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 17 | $\frac{4}{x^2-2x-8} + \frac{5}{x^2+5x+6}$ | $\frac{9x-8}{(x-4)(x+2)(x+3)}$ | | 18 | $\frac{x-2}{x^2+2x-8} + \frac{2}{x+4}$ | $\frac{3}{x+4}$ | | 19 | $\frac{x+3}{x^2-1} + \frac{1}{x+1}$ | $\frac{2}{x-1}$ | | 20 | $\frac{2x+1}{x^2+x} - \frac{1}{x+1}$ | $\frac{1}{x}$ | | 21 | $\frac{x+4}{x^2+5x} + \frac{x+1}{x^2+6x+5}$ | $\frac{2(x+2)}{x(x+5)}$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Rational Expressions" ] }

No.	Problem	Answer
18	\(\frac{x-2}{x^2+2x-8} + \frac{2}{x+4}\)	\(\frac{3}{x+4}\)
19	\(\frac{x+3}{x^2-1} + \frac{1}{x+1}\)	\(\frac{2}{x-1}\)
20	\(\frac{2x+1}{x^2+x} - \frac{1}{x+1}\)	\(\frac{1}{x}\)
21	\(\frac{x+4}{x^2+5x} + \frac{x+1}{x^2+6x+5}\)	\(\frac{2(x+2)}{x(x+5)}\)

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QUESTION IMAGE

Question

Explanation:

Factor denominators and find common denominators

Rewrite with common denominators and combine numerators

Simplify final expressions

Answer: