Question

Use the factor theorem to find all real zeros for the given polynomial and one of it's factors.Polynomial: f(x)=2x^3-9x^2+13x-6 Factor: x-1List the zero's from smallest to largest. If a zero is not an integer write it as a fraction.The zeros are Answer and Answer

Answer 1

Factor theorem

States that if f(a) = 0 for a polynomial, then (x-a) is a factor for the polynomial f(x).

• Polynomial given

`f(x)=2x^3-9x^2+13x-6`

and we are said that the factor is (x-1). Thus, we have to evaluate f(1):

`f(1)=2(1)^3-9(1)^2+13(1)-6`
`f(1)=2-9+13-6`
`f(1)=0`

Therefore, x = 1 is a zero.

To find the other zeros, we have to set the equation to 0 and factor it:

`0=2x^3-9x^2+13x-6`

Factoring we get:

`(x-2)\cdot(x-1)\cdot(2x-3)=0`

As we already know that (x -1) is a factor, we have to try with the others:

• (x-2)

`f(2)=2(2)^3-9(2)^2+13(2)-6`
`f(2)=16-36+26-6`
`f(2)=0`

Therefore, x = 2 is a zero.

• (2x-3)

`x=(3)/(2)`
`f((3)/(2))=2((3)/(2))^3-9((3)/(2))^2+13((3)/(2))-6`
`f((3)/(2))=(27)/(4)-(81)/(4)^{}+(39)/(2)-6`
`f((3)/(2))=(27)/(4))^{}-(81)/(4)^{}+(39)/(2)-6`
`f((3)/(2))=0`

Answer: 1, 2 and 3/2.