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