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=\frac{3}{2}$$f(\frac{3}{2})=2(\frac{3}{2})^3-9(\frac{3}{2})^2+13(\frac{3}{2})-6$$f(\frac{3}{2})=\frac{27}{4}-\frac{81}{4}^{}+\frac{39}{2}-6$$f(\frac{3}{2})=\frac{27}{4})^{}-\frac{81}{4}^{}+\frac{39}{2}-6$$f(\frac{3}{2})=0$

Answer: 1, 2 and 3/2.