Question

if 2 + sqrt(3) is a polynomial root, name another root of the polynomial, and explain how you know it must also be a root

Answer 1

Answer:

The another root of the polynomial $x^2-4x+1=0$  is $2-\sqrt{3}$

Step-by-step explanation:

let $x=2+\sqrt{3}$ is a polynomial root.

Subtracting both sides by 2, we get

$x-2=\sqrt{3}$

Squaring both sides, we get

$(x-2)^2=(\sqrt{3})^2$

$x^2+4-4x=3$

On solving we get,

$x^2-4x+1=0$

Using quadratic formula:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

If one of the root of the polynomial is $2+\sqrt{3}$ ; then another root by quadratic formula is $2-\sqrt{3}$

or else we can say that irrational roots will always come in conjugate pairs.

Sso, if $2+\sqrt{3}$is a root, then $2-\sqrt{3}$ is also a root.

Answer 2

We are given the root of a polynomial root 2 + sqrt(3) in which we are asked in the problem to determine the other root of the polynomial. SInce this is an irrational number, then the other root should be the conjugate of that number.Hence the answer to this problem should be 2 - sqrt 3.