Sure, let's go step by step explaining the logic behind this.
1. Let's start by recalling what a polynomial is. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
2. Polynomial equations can have roots. A root of a polynomial is a number 'x' such that when we substitute 'x' in the place of the variable in the polynomial, the value of the polynomial becomes zero.
3. We are given that a polynomial has one root in the form "a - sqrt(b)". Let's denote this root as x1. So, x1 = a - sqrt(b).
4. In polynomial equations, especially quadratic equations, if there's one root of the form "a - sqrt(b)", there's generally another root of the form "a + sqrt(b)". This happens due to the symmetry property of polynomials.
5. Hence, we can say that the other root of the polynomial is "a + sqrt(b)". Let's denote this root as x2. So, x2 = a + sqrt(b).
6. Both x1 and x2 satisfy the polynomial equation, meaning when we substitute x1 and x2 into our polynomial equation, the result is zero.
7. Therefore, we can conclude that "If a polynomial has one root in the form a - sqrt(b), it has a second root in the form a + sqrt(b)".