Answer:
The Factored form is
[tex]f(x)=(x-1)(x-3-\sqrt{13} )(x-3+\sqrt{13} )[/tex]
Step-by-step explanation:
Given;
[tex]f(x)=x^{3} -7\times x^{2} +2x+4[/tex]
To solve for x we factorize the right side of
[tex]f(x)=x^{3} -7\times x^{2} +2x+4[/tex]
Let Substitute x for various values to check whether remainder is zero.
[tex]f(1)=1^{3} -7\times 1^{2} +2\times 1+4[/tex]
Hence[tex]x-1[/tex] is a factor of the function.
Do synthetic division to find the quotient
[tex]1[/tex] [tex]1[/tex] [tex]-7[/tex] [tex]2[/tex] [tex]4[/tex]
[tex]1[/tex] [tex]-6[/tex] [tex]-4[/tex]
_____________________
[tex]1[/tex] [tex]-6[/tex] [tex]-4[/tex] [tex]0[/tex]
We get remainder 0 and quotient as,
[tex]x^{2} -(6\times x)-4[/tex]
By using Quadratic Formula;
[tex]x=\frac{-b\pm\sqrt{b^{2}-4ac } }{2a}[/tex]
Plug 'a' , 'b' and 'c' value from [tex]x^{2} -(6\times x)-4[/tex] equation,
[tex]x=\frac{-(-6)\pm\sqrt{-6^{2}-(4\times1\times -4 )} }{2\times1}[/tex]
[tex]x=\frac{6\pm\sqrt{36+16 } }{2}[/tex]
[tex]x=\frac{6\pm\sqrt{52} }{2}[/tex]
[tex]x=3+\sqrt{13}[/tex] and [tex]x=3-\sqrt{13}[/tex]
The values of 'x' are [tex]1[/tex] , [tex]3+\sqrt{13}[/tex] and [tex]3-\sqrt{13}[/tex]
So the Factored form is
[tex]f(x)=(x-1)(x-3-\sqrt{13} )(x-3+\sqrt{13} )[/tex]
