Question

How do you divide `(3x ^ { 3} + 13x ^ { 2} - 28x + 14) \div ( 3x - 2)`?

Answer 1

`x^2+5x-6+2/(3x-2)`

Explanation:

`"one way is to use the divisor as a factor in the numerator"`

`"consider the numerator"`

`color(red)(x^2)(3x-2)color(magenta)(+2x^2)+13x^2-28x+14`

`=color(red)(x^2)(3x-2)color(red)(+5x)(3x-2)color(magenta)(+10x)-28x+14`

`=color(red)(x^2)(3x-2)color(red)(+5x)(3x-2)color(red)(-6)(3x-2)color(magenta)(-12)+14`

`=color(red)(x^2)(3x-2)color(red)(+5x)(3x-2)color(red)(-6)(3x-2)+2`

`"quotient "=color(red)(x^2+5x-6)," remainder "=2`

`rArr(3x^3+13x^2-28x+14)/(3x-2)`

`=x^2+5x-6+2/(3x-2)`

Answer 2

The quotient is `=x^2+5x-6` and the remainder is `=2`

Explanation:

Let's perform a long division

`3x-2` `color(white)(aaaa)` `|` `3x^3+13x^2-28x+14` `color(white)(aaaa)` `|` `x^2+5x-6`

`color(white)(aaaaaaaaaaa)` `3x^3-2x^2`

`color(white)(aaaaaaaaaaaaa)` `0+15x^2-28x`

`color(white)(aaaaaaaaaaaaaaa)` `+15x^2-10x`

`color(white)(aaaaaaaaaaaaaaaaa)` `+0-18x+14`

`color(white)(aaaaaaaaaaaaaaaaaaaaa)` `-18x+12`

`color(white)(aaaaaaaaaaaaaaaaaaaaaaa)` `-0+2`

`(3x^3+13x^2-28x+14)/(3x-2)=x^2+5x-6+2/(3x-2)`