Question

If a + b√r is in F(√r) and is a root of a polynomial with rational coefficients, then a - b√r is also a root because:

a) Conjugate roots property
b) Binomial theorem
c) Descartes' Rule of Signs
d) Fundamental Theorem of Algebra

Answer 1

The correct answer is a) Conjugate roots property, as it is relevant when a polynomial with real coefficients has non-real complex roots, implying that the conjugate of the root is also a root of the polynomial.

Step-by-step explanation:

If a + b√r is in F(√) and is a root of a polynomial with rational coefficients, then a - b√r is also a root. The correct answer is a) Conjugate roots property. This is because if a polynomial has real coefficients and a non-real complex root, then its conjugate is also a root. In the case of the expression a + b√r, which belongs to the field F(√), its conjugate would be a - b√r. This property holds because the coefficients of the polynomial are rational (hence real), and the complex conjugate pair will ensure that the polynomial has real coefficients when expanded.

The quadratic formula, -b ± √b² - 4ac over 2a, illustrates that for every positive square root there is a negative square root, providing us two roots of the quadratic equation ax²+bx+c = 0. The existence of conjugate roots ensures the symmetry of these roots with respect to the real axis when plotted in the complex plane.