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(3x)/((x-5)^(2))+(6y-9z)/((x+5))

3x(x−5)2+6y−9z(x+5) \frac{3 x}{(x-5)^{2}}+\frac{6 y-9 z}{(x+5)}

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Full solution

Q. 3x(x−5)2+6y−9z(x+5) \frac{3 x}{(x-5)^{2}}+\frac{6 y-9 z}{(x+5)}
  1. Identify Terms and Denominators: Identify the terms and common denominators to simplify the expression. (3x)/((x−5)2)+(6y−9z)/((x+5))(3x)/((x-5)^{2}) + (6y-9z)/((x+5))
  2. Find Common Denominator: Find a common denominator, which is (x−5)2(x+5)(x-5)^2(x+5).3x(x−5)2⋅x+5x+5+6y−9z(x+5)⋅(x−5)2(x−5)2\frac{3x}{(x-5)^{2}} \cdot \frac{x+5}{x+5} + \frac{6y-9z}{(x+5)} \cdot \frac{(x-5)^2}{(x-5)^2}
  3. Multiply Numerators: Multiply the numerators by the appropriate expressions.\newline(3x)(x+5)/((x−5)2(x+5))+(6y−9z)(x−5)2/((x+5)(x−5)2)(3x)(x+5)/((x-5)^2(x+5)) + (6y-9z)(x-5)^2/((x+5)(x-5)^2)
  4. Expand Numerators: Expand the numerators.\newline3x(x+5)+(6y−9z)(x−5)23x(x+5) + (6y-9z)(x-5)^2\newline= 3x2+15x+(6y−9z)(x2−10x+25)3x^2 + 15x + (6y-9z)(x^2 - 10x + 25)
  5. Distribute Second Term: Distribute (6y−9z)(6y-9z) in the second term.\newline3x2+15x+6yx2−60yx+150y−9zx2+90zx−225z3x^2 + 15x + 6yx^2 - 60yx + 150y - 9zx^2 + 90zx - 225z
  6. Combine Like Terms: Combine like terms.\newline(3x2+6yx2−9zx2)+(15x−60yx+90zx)+(150y−225z)(3x^2 + 6yx^2 - 9zx^2) + (15x - 60yx + 90zx) + (150y - 225z)\newline= (3+6y−9z)x2+(−45y+75z)x+(150y−225z)(3 + 6y - 9z)x^2 + (-45y + 75z)x + (150y - 225z)

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