Respuesta :
Answer:
[tex]a - \sqrt{6}[/tex]
Step-by-step explanation:
Polynomial is a name made of two terms: poly and nomial where poly means many and nomial means terms. Thus, polynomial can be defined as an expression that is a sum of many terms expressed different powers of same variable.
For example: [tex]p( x) = 4x^{2} +6x +2[/tex] is an example of a polynomial.
To find roots of a polynomial, we equate p(x ) to 0 i.e. [tex]p(x)=0[/tex].
Whenever the roots are in radical form, it implies that they will occur as conjugates.
Conjugates means that if one of the root of an equation is [tex]a + \sqrt{b}[/tex], the other root will be [tex]a - \sqrt{b}[/tex]. To show that this is true and that the second root is of form [tex]a - \sqrt{b}[/tex] , we create a polynomial from the factors.
Factors are as follows: [tex]x - (a + \sqrt{6}) [/tex] and [tex]x- (a - \sqrt{6})[/tex]
Polynomial [tex] p(x) = (x - (a + \sqrt{6})) *( x- (a - \sqrt{6}))[/tex]
[tex]p(x)= x^{2} - x(a-\sqrt{6}) - x( a + \sqrt{6}) + (a+\sqrt{6})( a- \sqrt{6})[/tex]
[tex]p(x)= x^{2} - 2ax + x \sqrt{6}) - x \sqrt{6}) + a^{2} - 6[/tex]
[tex]p(x)= x^{2} - 2ax + a^{2} - 6[/tex]
which is an quadratic equation.
Now if we try to solve this equation by using the quadratic formula we get:
[tex]x = 1/2 [ 2a + \sqrt{4 a^{2}- 4( a^{2}- 6)}][/tex] and [tex]1/2 [ 2a - \sqrt{4 a^{2}- 4( a^{2}- 6)}][/tex]
[tex]x = a + (1/2) * \sqrt{24}[/tex] and [tex]x = a - (1/2) * \sqrt{24}[/tex]
[tex]x = a + \sqrt{6}[/tex] and [tex]x = a - \sqrt{6}[/tex]
Thus we get square roots of form [tex]a + \sqrt{6}[/tex] and [tex]a - \sqrt{6}[/tex].
