Question

What is the completely factored form of f(x)=x3−7x2+2x+4?

Answer 1

The Factored form is

`f(x)=(x-1)(x-3-√(13) )(x-3+√(13) )`

Explanation:

Given;

`f(x)=x^(3) -7* x^(2) +2x+4`

To solve for x we factorize the right side of

`f(x)=x^(3) -7* x^(2) +2x+4`

Let Substitute x for various values to check whether remainder is zero.

`f(1)=1^(3) -7* 1^(2) +2* 1+4`

Hence
`x-1` is a factor of the function.

Do synthetic division to find the quotient

`1`
`1`
`-7`
`2`
`4`

`1`
`-6`
`-4`

_____________________

`1`
`-6`
`-4`
`0`

We get remainder 0 and quotient as,

`x^(2) -(6* x)-4`

By using Quadratic Formula;

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

Plug 'a' , 'b' and 'c' value from
`x^(2) -(6* x)-4` equation,

`x=\frac{-(-6)\pm\sqrt{-6^(2)-(4*1* -4 )} }{2*1}`

`x=(6\pm√(36+16 ) )/(2)`

`x=(6\pm√(52) )/(2)`

`x=3+√(13)` and
`x=3-√(13)`

The values of 'x' are
`1` ,
`3+√(13)` and
`3-√(13)`

So the Factored form is

`f(x)=(x-1)(x-3-√(13) )(x-3+√(13) )`