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(x^(3)-27)/(3x^(2)-24 x+45)÷(x^(2)+3x+9)/(x^(2)-25)

66. x3−273x2−24x+45÷x2+3x+9x2−25 \frac{x^{3}-27}{3 x^{2}-24 x+45} \div \frac{x^{2}+3 x+9}{x^{2}-25}

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Full solution

Q. 66. x3−273x2−24x+45÷x2+3x+9x2−25 \frac{x^{3}-27}{3 x^{2}-24 x+45} \div \frac{x^{2}+3 x+9}{x^{2}-25}
  1. Factor Numerator and Denominator: Factor the numerator and denominator of the first fraction.\newline(x3−27)(x^3 - 27) can be factored as (x−3)(x2+3x+9)(x - 3)(x^2 + 3x + 9).\newline(3x2−24x+45)(3x^2 - 24x + 45) can be factored as 3(x2−8x+15)3(x^2 - 8x + 15), which further factors into 3(x−3)(x−5)3(x - 3)(x - 5).
  2. Factor Second Fraction: Factor the numerator and denominator of the second fraction.\newline(x2+3x+9)(x^2 + 3x + 9) is already in its simplest form and cannot be factored further.\newline(x2−25)(x^2 - 25) can be factored as (x−5)(x+5)(x - 5)(x + 5).
  3. Rewrite as Multiplication: Rewrite the division as multiplication by the reciprocal of the second fraction.\newlineThe expression becomes (x−3)(x2+3x+9)3(x−3)(x−5)×(x−5)(x+5)x2+3x+9\frac{(x - 3)(x^2 + 3x + 9)}{3(x - 3)(x - 5)} \times \frac{(x - 5)(x + 5)}{x^2 + 3x + 9}.
  4. Cancel Common Factors: Cancel out common factors from the numerator and denominator.\newlineThe (x−3)(x - 3) terms cancel, and one (x−5)(x - 5) term cancels.\newlineThe expression simplifies to x2+3x+93(x+5)\frac{x^2 + 3x + 9}{3(x + 5)}.
  5. Simplify Expression: Simplify the expression.\newlineThe final simplified form is (x2+3x+9)/(3x+15)(x^2 + 3x + 9)/(3x + 15).

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