Question

Y raised to 3 subtracted by y raised to 2 subtracted by 6y subtracted by 4 divided by y+1

Answer 1

The expression is
`(y^(3)-y^(2)-6y-4)/(y+1)=y^(2)-2y-4`

For
`y^(2)-2y-4=0` Factors are
`(1+√(5))` ,
`(1-√(5))`

Therefore
`y^(3)-y^(2)-6y-4y=(y+1)(1+√(5))(1-√(5))`

Explanation:

Given expression can be written as
`(y^(3)-y^(2)-6y-4)/(y+1)`

By using synthetic division we can find factors of given expression

1 -1 -6 -4

-1 | 0 -1 2 4

| _____________________

1 0 -4 0

Therefore quadratic equation is
`y^2-2y-4=0`

Here a=1 , b=-2 and c=-4

`y=\frac{-b\pm \sqrt{b^(2)-4ac}}{2a}`

`=\frac{-(-2)\pm \sqrt{(-2)^(2)-4(1)(-4)}}{2(1)}`

`=(2\pm √(4+16))/(2)`

`=(2\pm √(20))/(2)`

`=(2\pm √(4* 5))/(2)`

`=(2\pm 2√(5))/(2)`

`=(2(1\pm √(5)))/(2)`

`=1\pm √(5)`

Therefore
`y=1+√(5)` and
`y=1-√(5)`

The expression is
`(y^(3)-y^(2)-6y-4)/(y+1)=y^(2)-2y-4`

For
`y^(2)-2y-4=0` Factors are
`(1+√(5))` ,
`(1-√(5))`

Therefore
`y^(3)-y^(2)-6y-4y=(y+1)(1+√(5))(1-√(5))`