Question

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

Answer 1

Answer:

The Factored form is

$f(x)=(x-1)(x-3-\sqrt{13} )(x-3+\sqrt{13} )$

Step-by-step explanation:

Given;

$f(x)=x^{3} -7\times x^{2} +2x+4$

To solve for x  we factorize the right side  of

$f(x)=x^{3} -7\times x^{2} +2x+4$

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

$f(1)=1^{3} -7\times 1^{2} +2\times 1+4$

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,

$x^{2} -(6\times x)-4$

By using Quadratic Formula;

$x=\frac{-b\pm\sqrt{b^{2}-4ac } }{2a}$

Plug 'a' , 'b' and 'c' value from $x^{2} -(6\times x)-4$ equation,

$x=\frac{-(-6)\pm\sqrt{-6^{2}-(4\times1\times -4 )} }{2\times1}$

$x=\frac{6\pm\sqrt{36+16 } }{2}$

$x=\frac{6\pm\sqrt{52} }{2}$

$x=3+\sqrt{13}$ and $x=3-\sqrt{13}$

The values of 'x' are $1$ , $3+\sqrt{13}$ and   $3-\sqrt{13}$

So the Factored form is

$f(x)=(x-1)(x-3-\sqrt{13} )(x-3+\sqrt{13} )$