Question

If a polynomial function f(x) has roots 3+root5 and -6, what must be a factor of f(x)? (X+(3-root5) (x-(3-root5)) (x+(5+root3)) (x-(5-root3))

Answer 1

`(x-(3+\sqrt5))` or
`(x-3-\sqrt5)` is a factor of the given polynomial.

Explanation:

Let us learn the concept with an example first.

Let the polynomial be a quadratic function
`g(x)`.

`g(x) = x^(2) -5x+6`

The roots of
`g(x)` are 2 and 3.

Putting
`x= 2\ in \ g(x)`

`2^2-5* 2+6 = 4-10+6 =0`

Putting
`x= 3\ in \ g(x)`

`3^2-5* 3+6 = 9-15+6 =0`

Putting x = 2 or x = 3, g(x) = 0
`\therefore` The roots of equation g(x) are 2 and 3.

Now, let us try to factorize g(x):

`x^(2) -2x-3x+6\\\Rightarrow x(x -2)-3(x-2)\\\Rightarrow (x-3)(x-2)`

so, the equation can be written as:

`g(x) = x^(2) -5x+6=(x-3)(x-2)` where 3 and 2 are the roots of equation.

The factors are (x-3) and (x-2).

`\therefore` for the polynomial f(x) which has roots
`3+\sqrt5\ and\ -6` will have a factor:

`(x-(3+\sqrt5))` or
`(x-3-\sqrt5)`