# Factor the polynomial function to determine the x intercepts and the y intercept.  Enter all of the x-intercepts from smallest to largest.  Enter a rational answer as a fraction.          ...

Factor the polynomial function to determine the x intercepts and the y intercept.  Enter all of the x-intercepts from smallest to largest.  Enter a rational answer as a fraction.

y = 4x^3 - 12x^2 - 9x + 27

llltkl | College Teacher | (Level 3) Valedictorian

Posted on

The given polynomial is:

`y = 4x^3 - 12x^2 - 9x + 27`

By factoring, we get

`y = 4x^2(x-3) -9(x - 3)`

`=(x-3)(4x^2-9)`

`=(x-3)((2x)^2-3^2) `

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

For x-intercepts, put y=0

So, `(x-3)(2x-3)(2x+3)=0`

Setting each factor equal to zero results in three solutions for x, viz.

`(x-3)=0 rArr x=3`

`(2x-3)=0 rArr x=3/2`

`(2x+3)=0 rArr x=-3/2`

Hence the x-intercepts of the given polynomial are -3/2, 3/2 and 3.

For y-intercept, put x=0

So, `y = 4*0^3 - 12*0^2 - 9*0 + 27`

`rArr y=27`

Therefore, the y-intercept of the given polynomial is 27.