Question

How do you divide `\frac { x ^ { 2} + 3x + 2} { 3x ^ { 2} + x - 2} \div ( x + 2)`?

Answer 1

`(3x-2)`

Explanation:

When you are simplifying algebraic fractions, factorise first wherever possible.

`(x^2+3x+2)/(3x^2+x-2) div (x+2)/1`

`((x+2)(x+1))/((3x-2)(x+1)) xx 1/((x+2))" "larr` cancel common factors

`=((x+2)(x+1))/((3x-2)(x+1)) xx 1/((x+2))`

`=(cancel((x+2))cancel((x+1)))/((3x-2)cancel((x+1))) xx 1/cancel((x+2))`

`=(3x-2)`

Answer 2

Lets begin by factorizing each quadratic trinomial,

`color(red)(x^2+3x+2`

`=> x^2 + x + 2x+ 2`

`=> x(x+1) + 2(x+1)`

`=> color(red)((x+ 2)(x+1)`

`color(magenta)(3x^2+x−2`

`=>3x^2+3x-2x−2`

`=>3x(x+1)-2(x+1)`

`=> color(magenta)((3x-2)(x+1)`

`"Now, according to the question, we need to find,"`

`(color(red)(x^2+3x+2))/(color(magenta)(3x^2+x−2)) xx 1/(x+2)`

`=> color(red)(cancel((x+ 2))cancel((x+1)))/ color(magenta)((3x-2)cancel((x+1)))xx 1/(cancel((x+2))`

`=> 1/(3x-2`