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QUESTION IMAGE

calculate and simplify the following rational expressions. 1) \\(\\frac…

Question

calculate and simplify the following rational expressions.

  1. \\(\frac{6x}{3} + \frac{7}{15x + 3}\\)
  1. \\(\frac{5v}{v - 3} + \frac{5}{v + 6}\\)
  1. \\(\frac{4x}{x^2 + 4x - 5} - \frac{5}{4}\\)
  1. \\(\frac{2}{x + 3} - \frac{6x}{2x + 1}\\)
  1. \\(\frac{4x}{x + 3} - \frac{4x}{x + 6}\\)
  1. \\(\frac{2x}{3x + 3} - \frac{2}{x + 5}\\)
  1. \\(\frac{6}{x - 2} + \frac{6}{x + 1}\\)
  1. \\(\frac{v - 2}{3v^4 - 15v^3 - 18v^2} + 3v\\)

Explanation:

Find common denominators

We identify the denominators for each rational expression to find their least common denominator (LCD).

  • Question 1: Denominators are \(3\) and \(15x+3 = 3(5x+1)\). The LCD is \(3(5x+1)\).
  • Question 2: Denominators are \(v-3\) and \(v+6\). The LCD is \((v-3)(v+6)\).
  • Question 3: Denominators are \(x^2+4x-5 = (x+5)(x-1)\) and \(4\). The LCD is \(4(x+5)(x-1)\).
  • Question 4: Denominators are \(x+3\) and \(2x+1\). The LCD is \((x+3)(2x+1)\).
  • Question 5: Denominators are \(x+3\) and \(x+6\). The LCD is \((x+3)(x+6)\).
  • Question 6: Denominators are \(3x+3 = 3(x+1)\) and \(x+5\). The LCD is \(3(x+1)(x+5)\).
  • Question 7: Denominators are \(x-2\) and \(x+1\). The LCD is \((x-2)(x+1)\).
  • Question 8: Denominators are \(3v^4-15v^3-18v^2 = 3v^2(v^2-5v-6) = 3v^2(v-6)(v+1)\) and \(1\). The LCD is \(3v^2(v-6)(v+1)\).

Rewrite with common denominators

We adjust the numerators by multiplying them by the missing factors from the LCD.

  • Question 1: \(\frac{6x}{3} + \frac{7}{15x+3} = \frac{2x(15x+3) + 7}{15x+3}\) or simplify first: \(2x + \frac{7}{3(5x+1)} = \frac{2x(15x+3) + 7}{3(5x+1)}\).
  • Question 2: \(\frac{5v(v+6) + 5(v-3)}{(v-3)(v+6)}\).
  • Question 3: \(\frac{4x(4) - 5(x^2+4x-5)}{4(x^2+4x-5)}\).
  • Question 4: \(\frac{2(2x+1) - 6x(x+3)}{(x+3)(2x+1)}\).
  • Question 5: \(\frac{4x(x+6) - 4x(x+3)}{(x+3)(x+6)}\).
  • Question 6: \(\frac{2x(x+5) - 2(3x+3)}{3(x+1)(x+5)}\).
  • Question 7: \(\frac{6(x+1) + 6(x-2)}{(x-2)(x+1)}\).
  • Question 8: \(\frac{v-2 + 3v(3v^2(v-6)(v+1))}{3v^2(v-6)(v+1)}\).

Simplify the numerators

We expand and combine like terms in the numerators.

  • Question 1: \(2x + \frac{7}{15x+3} = \frac{2x(15x+3)+7}{15x+3} = \frac{30x^2+6x+7}{3(5x+1)}\).
  • Question 2: \(5v^2+30v + 5v-15 = 5v^2+35v-15 = 5(v^2+7v-3)\).
  • Question 3: \(16x - 5x^2 - 20x + 25 = -5x^2-4x+25\).
  • Question 4: \(4x+2 - 6x^2-18x = -6x^2-14x+2 = -2(3x^2+7x-1)\).
  • Question 5: \(4x^2+24x - (4x^2+12x) = 12x\).
  • Question 6: \(2x^2+10x - 6x-6 = 2x^2+4x-6 = 2(x^2+2x-3) = 2(x+3)(x-1)\).
  • Question 7: \(6x+6 + 6x-12 = 12x-6 = 6(2x-1)\).
  • Question 8: \(v-2 + 9v^3(v^2-5v-6) = v-2 + 9v^5-45v^4-54v^3 = 9v^5-45v^4-54v^3+v-2\).

Final algebraic simplification

We check for any common factors between the simplified numerators and denominators to reduce the fractions.

  • Question 1: \(\frac{30x^2+6x+7}{3(5x+1)}\) (No common factors).
  • Question 2: \(\frac{5(v^2+7v-3)}{(v-3)(v+6)}\) (No common factors).
  • Question 3: \(\frac{-5x^2-4x+25}{4(x+5)(x-1)}\) (No common factors).
  • Question 4: \(\frac{-2(3x^2+7x-1)}{(x+3)(2x+1)}\) (No common factors).
  • Question 5: \(\frac{12x}{(x+3)(x+6)}\) (No common factors).
  • Question 6: \(\frac{2(x+3)(x-1)}{3(x+1)(x+5)}\) (No common factors).
  • Question 7: \(\frac{6(2x-1)}{(x-2)(x+1)}\) (No common…

Answer:

No.Answer
2\(\frac{5(v^2+7v-3)}{(v-3)(v+6)}\)
3\(\frac{-5x^2-4x+25}{4(x+5)(x-1)}\)
4\(\frac{-2(3x^2+7x-1)}{(x+3)(2x+1)}\)
5\(\frac{12x}{(x+3)(x+6)}\)
6\(\frac{2(x+3)(x-1)}{3(x+1)(x+5)}\)
7\(\frac{6(2x-1)}{(x-2)(x+1)}\)
8\(\frac{9v^5-45v^4-54v^3+v-2}{3v^2(v-6)(v+1)}\)