Final answer:
To find the additional roots of a polynomial with given complex roots, you must include the conjugate of those roots. The quadratic formula is used for equations of the form ax² + bx + c = 0. Complex roots appear in conjugate pairs if the polynomial has real coefficients.
Step-by-step explanation:
The student's question is about finding the additional roots of a polynomial function P(x) with rational coefficients, given that it already has some roots. The approach to solving this involves using the quadratic formula. Given the nature of polynomial functions, if the coefficients are rational and we have an irrational or complex root, then its conjugate must also be a root of the polynomial.
Therefore, if the roots include complex numbers or radicals, we must also include their conjugates as roots. A general quadratic equation is of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants. The roots of this equation can be found using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a.
It's important to note that for complex roots, if 'b² - 4ac' is negative, the solutions will involve i, which is the imaginary unit. When solving a quadratic equation with complex roots, the two roots will be of the form a + bi and a - bi. In this particular scenario, given that 17 and -81 are roots, if these are not rational numbers or do not fit the rational coefficients of the polynomial, we may need more information to provide additional roots, or it may be a simple typo that we should ignore in the calculation.
However, assuming that '17 + 16i' and '-81i' are the roots mentioned, we can infer that '17 - 16i' and '+81i' are also roots due to the nature of complex roots occurring in conjugate pairs in polynomials with real coefficients.