Answer:
[tex](x-(3+\sqrt5))[/tex] or [tex](x-3-\sqrt5)[/tex] is a factor of the given polynomial.
Step-by-step explanation:
Let us learn the concept with an example first.
Let the polynomial be a quadratic function [tex]g(x)[/tex].
[tex]g(x) = x^{2} -5x+6[/tex]
The roots of [tex]g(x)[/tex] are 2 and 3.
Putting [tex]x= 2\ in \ g(x)[/tex]
[tex]2^2-5\times 2+6 = 4-10+6 =0[/tex]
Putting [tex]x= 3\ in \ g(x)[/tex]
[tex]3^2-5\times 3+6 = 9-15+6 =0[/tex]
Putting x = 2 or x = 3, g(x) = 0 [tex]\therefore[/tex] The roots of equation g(x) are 2 and 3.
Now, let us try to factorize g(x):
[tex]x^{2} -2x-3x+6\\\Rightarrow x(x -2)-3(x-2)\\\Rightarrow (x-3)(x-2)[/tex]
so, the equation can be written as:
[tex]g(x) = x^{2} -5x+6=(x-3)(x-2)[/tex] where 3 and 2 are the roots of equation.
The factors are (x-3) and (x-2).
[tex]\therefore[/tex] for the polynomial f(x) which has roots [tex]3+\sqrt5\ and\ -6[/tex] will have a factor:
[tex](x-(3+\sqrt5))[/tex] or [tex](x-3-\sqrt5)[/tex]
