Question

A polynomial function has α as a root. Which of the following must also be a root of the function:

A. Apply the Factor Theorem to identify additional roots.
B. Evaluate the complex conjugate of α as a possible root.
C. Analyze the relationship between roots and factors of a polynomial.
D. Discuss the implications of the given root for the polynomial's factorization.

Answer 1

To find additional roots of a polynomial function with a given root, we evaluate the complex conjugate of the given root.

Step-by-step explanation:

To determine which of the given options must also be a root of the polynomial function with α as a root, we can use the fact that complex roots appear in conjugate pairs. This means that if α is a root, then its complex conjugate, denoted as α*, must also be a root.

Therefore, the correct option is B. Evaluate the complex conjugate of α as a possible root.

For example, if α = 2 + 3i, then α* = 2 - 3i, and both α and α* would be roots of the polynomial function.

Answer 2

The Factor Theorem can be used to identify additional roots of a polynomial function.

Step-by-step explanation:

To identify additional roots of a polynomial function, we can use the Factor Theorem. The Factor Theorem states that if a polynomial function has a root α, then (x - α) is a factor of the polynomial. So, if α is a root of the polynomial, then (x - α) must also be a factor, and therefore, a root as well.

For example, if α = 2 is a root of the polynomial, then (x - 2) is a factor of the polynomial, and x = 2 is another root. Similarly, if α = -3 is a root, then (x + 3) is a factor and x = -3 is another root.