How do you simplify (3x)/(x^2-x-12) - (x-1)/(x^2+6x+9) + (x-6)/(2x+6) ?

2 Answers
Jun 8, 2016

(3x)/(x^2-x-12)-(x-1)/(x^2+6x+9)+(x-6)/(2x+6)=(x^3-3x^2+22x+64)/(2(x+3)(x+3)(x-4))

Explanation:

For simplifying this, we first need to factorize denominators (as numerators are already in simplest terms).

x^2-x-12=x^2-4x+3x-12=x(x-4)+3(x-4)=(x+3)(x-4)

x^2+6x9=x^2+3x+3x+9=x(x+3)+3(x+3)=(x+3)(x+3)=(x+3)^2

2x+6=2(x+3)

And LCD of three denominators is 2(x+3)(x+3)(x-4)

Hence (3x)/(x^2-x-12)-(x-1)/(x^2+6x+9)+(x-6)/(2x+6)

= (3x)/((x+3)(x-4))-(x-1)/(x+3)^2+(x-6)/(2(x+3))

= (3x xx2xx(x+3)-(x-1)xx2(x-4)+(x-6)(x+3)(x-4))/(2(x+3)(x+3)(x-4))

= ((6x^2+18x)-2(x^2-5x+4)+(x-6)(x^2-x-12))/(2(x+3)(x+3)(x-4))

= ((6x^2+18x)-2(x^2-5x+4)+(x^3-x^2-12x-6x^2+6x+72))/(2(x+3)(x+3)(x-4))

= (x^3-3x^2+22x+64)/(2(x+3)(x+3)(x-4))

Jun 8, 2016

(x^3-3x^2+22x+64)/(2(x-4)(x+3)^2)

Explanation:

Look for common factors. So we need to investigate factorisation of the denominators.

x^2-x-12 = (x+3)(x-4)

x^2+6x+9=(x+3)(x+3)

2x+6=2(x+3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Factor out (x+3) from the denominators giving:

1/(x+3) [color(white)(.)(3x)/(x-4)-(x-1)/(x+3)+(x-6)/2color(white)(.)]color(red)("... Eqn (1)")

Using a common denominator of 2(x-4)(x+3)

'.................................................................................................................
Consider (3x)/(x-4) -> (3x xx2xx(x+3))/(2(x-4)(x+3)) = (6x^2+18x)/(2(x-4)(x+3))
,...................................................................................................................

Consider -(x-1)/(x+3)-> ((x-1)xx2xx(x-4))/(2(x-4)(x+3)) = -(2x^2-10x+8)/(2(x-4)(x+3)) .
'.....................................................................................................................

Consider (x-6)/2 -> ((x-6)(x-4)(x+3))/(2(x-4)(x+3))

=(x^3-7x^2-6x+72)/(2(x-4)(x+3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Putting it all together")

1/(x+3)[(x^3-3x^2+22x+64)/(2(x-4)(x+3))]

(x^3-3x^2+22x+64)/(2(x+3)(x-4)(x+3))

Factoring this further

((x+2)(x^2-5x+32))/(2(x-4)(x+3)^2) larr" don't think this helps much"