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Fully simplify the expression below and write your answer as a single fraction.

(2x^(2)-14 x)/(x^(5)-6x^(4)-7x^(3))*(x^(2)-9x-10)/(x^(2)-16 x+60)
Answer:

Fully simplify the expression below and write your answer as a single fraction.\newline2x214xx56x47x3x29x10x216x+60 \frac{2 x^{2}-14 x}{x^{5}-6 x^{4}-7 x^{3}} \cdot \frac{x^{2}-9 x-10}{x^{2}-16 x+60} \newlineAnswer:

Full solution

Q. Fully simplify the expression below and write your answer as a single fraction.\newline2x214xx56x47x3x29x10x216x+60 \frac{2 x^{2}-14 x}{x^{5}-6 x^{4}-7 x^{3}} \cdot \frac{x^{2}-9 x-10}{x^{2}-16 x+60} \newlineAnswer:
  1. Identify Given Expression: Identify the given expression and look for common factors in the numerator and denominator that can be simplified.\newlineThe expression is: (2x214x)/(x56x47x3)×(x29x10)/(x216x+60)(2x^2 - 14x) / (x^5 - 6x^4 - 7x^3) \times (x^2 - 9x - 10) / (x^2 - 16x + 60)
  2. Factor Numerator and Denominator: Factor the first numerator and the first denominator.\newline2x214x2x^2 - 14x can be factored as 2x(x7)2x(x - 7).\newlinex56x47x3x^5 - 6x^4 - 7x^3 can be factored as x3(x26x7)x^3(x^2 - 6x - 7).
  3. Factor Second Numerator and Denominator: Factor the second numerator and the second denominator. x29x10x^2 - 9x - 10 can be factored as (x10)(x+1)(x - 10)(x + 1). x216x+60x^2 - 16x + 60 can be factored as (x10)(x6)(x - 10)(x - 6).
  4. Rewrite with Factored Terms: Rewrite the expression with the factored terms.\newline2x(x7)x3(x26x7)(x10)(x+1)(x10)(x6)\frac{2x(x - 7)}{x^3(x^2 - 6x - 7)} \cdot \frac{(x - 10)(x + 1)}{(x - 10)(x - 6)}
  5. Cancel Common Factors: Cancel out the common factors in the numerator and denominator.\newlineThe (x10)(x - 10) terms cancel out, and one xx term from the first denominator cancels with the xx term in the first numerator.\newlineThe simplified expression is now: 2(x7)x2(x26x7)×x+1x6\frac{2(x - 7)}{x^2(x^2 - 6x - 7)} \times \frac{x + 1}{x - 6}
  6. Factor Remaining Quadratic: Factor the remaining quadratic in the denominator. x26x7x^2 - 6x - 7 can be factored as (x7)(x+1)(x - 7)(x + 1).
  7. Rewrite with Newly Factored Term: Rewrite the expression with the newly factored term. 2(x7)x2(x7)(x+1)x+1x6\frac{2(x - 7)}{x^2(x - 7)(x + 1)} \cdot \frac{x + 1}{x - 6}
  8. Cancel Common Factors: Cancel out the common factors in the numerator and denominator.\newlineThe (x7)(x - 7) and (x+1)(x + 1) terms cancel out.\newlineThe simplified expression is now: 2x2(x6)\frac{2}{x^2(x - 6)}
  9. Write Final Simplified Expression: Write the final simplified expression as a single fraction.\newlineThe final answer is: 2x36x2\frac{2}{x^3 - 6x^2}

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