Question

Find the rational zeros of the polynomial function. Then write each function in factored form.f(x)=6x^3-7x^2-9x-2

Answer 1

The given function is,

`6x^3-7x^2-9x-2`

polynomial equation with integer coefficient

`a_nx^n+a_(n-1)x^(n-1)+\ldots+a_0`
`\begin{gathered} ifa_{0\text{ }},a_1\text{ are integres and if the rational root exist we can find by } \\ \text{checking all the numbers produced by } \\ \pm(\: dividers\: of\: a_0)/(dividers\: of\: a_n) \\ \end{gathered}`

Here,

`a_0=2,\: \quad a_n=6`

The divisors of

`\begin{gathered} a_0\colon\quad 1,\: 2 \\ a_n\colon\quad 1,\: 2,\: 3,\: 6 \end{gathered}`

The rational roots are,

`\pm(1,\:2)/(1,\:2,\:3,\:6)`

validate the roots by plugging them into the function

`6x^3-7x^2-9x-2=0`

`x=2,x=-(1)/(2),x=-(1)/(3)`

The factor of the equation are,

`\begin{gathered} (x-2)\mleft(2x+1\mright)\mleft(3x+1\mright) \\ \end{gathered}`