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

Answer:

$(x-(3+\sqrt5))$ or $(x-3-\sqrt5)$ 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 $g(x)$.

$g(x) = x^{2} -5x+6$

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

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

$2^2-5\times 2+6 = 4-10+6 =0$

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

$3^2-5\times 3+6 = 9-15+6 =0$

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 $(x-3-\sqrt5)$