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

According to question 
root 1 = 2 + root 3

and as we know
that  two root of an equation are   b + root 4ac / 2a and b - root 4ac
Thus
Another root

Root 2 = 2 - root 3

Answer 2

Answer:

$\text{The other root is }2-\sqrt3$

Step-by-step explanation:

$\text{Given that }2 + \sqrt{3}\text{ is a polynomial root. }$

we have to find the another root of the polynomial.

$\text{Let }x=2+\sqrt3$

$x-2=\sqrt3$

$(x-2)^2=3$

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

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

which is the above polynomial

By quadratic formula

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}=\frac{4\pm \sqrt{(-4)^2-4(1)(1)}}{2}$

$x=\frac{4\pm \sqrt{12}}{2}=2\pm \sqrt3$

which are required roots.

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

$\text{If one of the root of the polynomial is }2+\sqrt3\text{ then another root is } 2-\sqrt3$

$\text{Hence, the other root is }2-\sqrt3$