How do you simplify #(x^3 - 9x) / (x^2 - 7x + 12)#?

Question

Genesis Allen · Accepted Answer

#(x(x+3))/(x-4)# #"factorise numerator and denominator"# #color(magenta)"factor numerator"# #"take out a "color(blue)"common factor "x# #=x(x^2-9)# #x^2-9" is a "color(blue)"difference of squares"# #"which factors in general as"# #•color(white)(x)a^2-b^2=(a-b)(a+b)# #"here "a=x" and "b=3# #rArrx^2-9=(x-3)(x+3)# #rArrx^3-9x=x(x-3)(x+3)larrcolor(red)"factorised form"# #color(magenta)"factor denominator"# #"the factors of + 12 which sum to - 7 are - 3 and - 4"# #rArrx^2-7x+12=(x-3)(x-4)larrcolor(red)"factored form"# #rArr(x^3-9x)/(x^2-7x+12)# #=(x(x-3)(x+3))/((x-3)(x-4))# #"cancel the "color(blue)"common factor "(x-3)# #=(xcancel((x-3))(x+3))/(cancel((x-3))(x-4))=(x(x+3))/(x-4)# #"with restriction "x!=4#

Nathan Axel · Answer

#(x(x+3))/(x-4#

#(x^3-9x)/(x^2-7x+12)# #color(teal)(=(x(x^2-9))/(x^2-3x-4x+12)#

#color(blue)(=(x(x+3)(x-3))/((x-3)(x-4))#

#color(magenta)(=(x(x+3)cancel((x-3)))/(cancel((x-3))(x-4))#

#color(green)(=(x(x+3))/(x-4#

Ans that's your answer.

P.S.: Isn't the solution #color(blue)c#olorful?

Eli Assouline · Answer

undefined To simplify the expression (x^3 - 9x) / (x^2 - 7x + 12), we can factor the numerator and denominator. The numerator can be factored as x(x^2 - 9), and the denominator can be factored as (x - 3)(x - 4). Canceling out the common factors, we get x(x - 3) / (x - 3)(x - 4). The (x - 3) terms cancel out, leaving us with x / (x - 4) as the simplified expression.