Question

If a polynomial has one root in the form a+square root of 6, it has a second root in the form of a_square root of b

Answer 1

`a - √(6)`

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:
`p( x) = 4x^(2) +6x +2` is an example of a polynomial.

To find roots of a polynomial, we equate p(x ) to 0 i.e.
`p(x)=0`.

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
`a + √(b)`, the other root will be
`a - √(b)`. To show that this is true and that the second root is of form
`a - √(b)` , we create a polynomial from the factors.

Factors are as follows:
`x - (a + √(6))` and
`x- (a - √(6))`

Polynomial
`p(x) = (x - (a + √(6))) *( x- (a - √(6)))`

`p(x)= x^(2) - x(a-√(6)) - x( a + √(6)) + (a+√(6))( a- √(6))`

`p(x)= x^(2) - 2ax + x √(6)) - x √(6)) + a^(2) - 6`

`p(x)= x^(2) - 2ax + a^(2) - 6`

which is an quadratic equation.

Now if we try to solve this equation by using the quadratic formula we get:

`x = 1/2 [ 2a + \sqrt{4 a^(2)- 4( a^(2)- 6)}]` and
`1/2 [ 2a - \sqrt{4 a^(2)- 4( a^(2)- 6)}]`

`x = a + (1/2) * √(24)` and
`x = a - (1/2) * √(24)`

`x = a + √(6)` and
`x = a - √(6)`

Thus we get square roots of form
`a + √(6)` and
`a - √(6)`.