Question

A polynomial function, f(x), with rational coefficients has √–2 and √3. the irrational conjugates theorem states that which of the following must also be a root of the function?

Answer 1

The irrational conjugates theorem demands that if a polynomial with rational coefficients has an irrational number as a root, then its irrational conjugate must also be a root. In this case, the additional roots must be –√–2 and –√3.

Step-by-step explanation:

The student has asked which other roots must be present if a polynomial function with rational coefficients has roots of √–2 and √3. The fact that the polynomial has rational coefficients points us to use the Irrational Conjugates Theorem, which states that if a polynomial has rational coefficients and an irrational number is a root, then the irrational conjugate of that number must also be a root.

Therefore, if √–2 is a root, then –√–2 must also be a root. Similarly, if √3 is a root, then –√3 must also be a root. These are known as complex conjugates and irrational conjugates respectively.

When dealing with quadratic equations or higher-order polynomials, recognizing these patterns is crucial when attempting to find all the roots of the function. For example, the solution of quadratic equations may involve complex or irrational numbers, and knowing such rules can greatly simplify the process of finding all possible roots.