How do you simplify #(2x^2+x-15)/(2x^2-11x-21)*(6x+9)div(2x-5)/(3x-21)#?

Question

Eva Adamson · Accepted Answer

#9(x+3)#

Given: #(2x^2+x-15)/(2x^2-11x-21) * (6x+9) -: (2x-5)/(3x-21)#

Change the division problem to multiplication by reciprocating the last rational function:

#(2x^2+x-15)/(2x^2-11x-21) * (6x+9) * (3x-21)/(2x-5)#

Make the second polynomial a fraction:

#(2x^2+x-15)/(2x^2-11x-21) * (6x+9)/1 * (3x-21)/(2x-5)#

Factor each polynomial:

#((x+3)(2x-5))/((2x+3)(x-7)) * (3(2x+3))/1 * (3(x-7))/(2x-5)#

Cancel all factors that occur in both the numerator & denominator:

#(color(red)(x+3)cancel(2x-5))/(cancel(2x+3)cancel(x-7)) * (color(red)(3)cancel(2x+3))/1 * (color(red)(3)cancel(x-7))/(cancel(2x-5))#

#= 9(x+3)#

Jace Anderson · Answer

undefined To simplify the expression (2x^2+x-15)/(2x^2-11x-21)*(6x+9)/(2x-5)/(3x-21), we can follow these steps:

1. Factorize the quadratic expressions in the numerators and denominators:
   (2x^2+x-15) = (2x-3)(x+5)
   (2x^2-11x-21) = (2x+3)(x-7)
   (6x+9) = 3(2x+3)
   (2x-5) = (2x-5)
   (3x-21) = 3(x-7)

2. Cancel out common factors between the numerators and denominators:
   (2x-3)(x+5)/(2x+3)(x-7) * 3(2x+3)/(2x-5)/(x-7)

3. Multiply the remaining factors together:
   (2x-3)(x+5) * 3(2x+3) / (2x+3)(x-7)(2x-5)

4. Cancel out common factors between the numerators and denominators:
   (2x-3)(x+5) * 3 / (x-7)(2x-5)

Therefore, the simplified expression is (2x-3)(x+5) * 3 / (x-7)(2x-5).