Question

How do you simplify and divide `(y^3+3y^2-5y-4)/(y+4)`?

Answer 1

The quotient is `=y^2-y-1`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `-4` `color(white)(aaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `3` `color(white)(aaaaa)` `-5` `color(white)(aaaa)` `-4`
`color(white)(aaaaaaaaaaaa)`_________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaa)` `-4` `color(white)(aaaaaa)` `4` `color(white)(aaaaaa)` `4`
`color(white)(aaaaaaaaaaaa)`________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaa)` `-1` `color(white)(aaaaa)` `-1` `color(white)(aaaaaa)` `color(red)(0)`

The remainder is `0` and the quotient is `=y^2-y-1`

`(y^3+3y^2-5y-4)/(y+4)=y^2-y-1`

Answer 2

`y^2-y-1`

Explanation:

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

`"consider the numerator"`

`color(red)(y^2)(y+4)color(magenta)(-4y^2)+3y^2-5y-4`

`=color(red)(y^2)(y+4)color(red)(-y)(y+4)color(magenta)(+4y)-5y-4`

`=color(red)(y^2)(y+4)color(red)(-y)(y+4)color(red)(-1)(y+4)color(magenta)(+4)-4`

`=color(red)(y^2)(y+4)color(red)(-y)(y+4)color(red)(-1)(y+4)+0`

`rArr(cancel((y+4))(color(red)(y^2-y-1)))/cancel((y+4))`

`=y^2-y-1larr" quotient"`