How do you simplify the rational expression: #(x+2)/(4x-8)(3x-9)/(x+4)(2x-21)/(x^2-x-6)#?

Question

Nathan Axel · Accepted Answer

#(6x-63)/(4 x^2+8 x-32)#

The idea is to factor out numbers is the first-degree polynomial (if possible), and to factor the highest-degree, finding their roots (if possible). So, let's work separately on the three pieces:

First fraction: Numerator: #x+2-># nothing to do; **Denominator: #4x-8-># can factor a #4#, obtaining #4(x-2)#.

Second fraction: Numerator: #3x-9-># can factor a #3#, obtaining #3(x-3)#; **Denominator: #x+4-># nothing to do.

First fraction: Numerator: #2x-21-># nothing to do; **Denominator: #x^2-x-6-># its roots are #3# and #-2#, so we can write it as #(x-3)(x+2)#.

Writing back the whole expression with this changes gives

#color(red)(cancel(x+2))/(4(x-2)) * (3color(blue)cancel((x-3)))/(x+4) * (2x-21)/(color(blue)cancel((x-3))color(red)cancel((x+2)))#

Julia Adams · Answer

undefined To simplify the rational expression (x+2)/(4x-8)(3x-9)/(x+4)(2x-21)/(x^2-x-6), you can follow these steps:

1. Factor all the denominators and numerators:
   (x+2)/(4x-8)(3x-9)/(x+4)(2x-21)/(x^2-x-6)
   = (x+2)/(4(x-2))(3(x-3))/(x+4)(2(x-7))/((x-3)(x+2))

2. Cancel out any common factors between the numerators and denominators:
   = 1/(4(x-2))(1)/(x+4)(2(x-7))/((x-3)(1))
   = 1/(4(x-2))/(x+4)(2(x-7))/(x-3)

3. Multiply the remaining factors together:
   = 1/(4(x-2)(x+4)(2(x-7))/(x-3)

Therefore, the simplified form of the rational expression is 1/(4(x-2)(x+4)(2(x-7))/(x-3).