Question

Patricia is studying a polynomial function f(x). Three given roots of f(x) = -11 - √2i, 3 + 4i, and 1. Patricia concludes that f(x) must be a polynomial degree 4. Which statements are true?

a) The degree of a polynomial is equal to the number of its roots.

b) The degree of a polynomial is determined by the highest power of the variable.

c) Patricia's conclusion is incorrect.

d) Both a and b.

Answer 1

Patricia's conclusion is incorrect because complex roots occur in conjugate pairs, suggesting that there should be at least two more roots than she accounted for. This makes the polynomial at least degree 5, not 4.

Step-by-step explanation:

The correct answers are b) The degree of a polynomial is determined by the highest power of the variable, and c) Patricia's conclusion is incorrect. Complex roots of polynomials come in conjugate pairs, so if the polynomial has a root at -11 - √2i, it should also have a root at -11 + √2i, which was not listed. Similarly, since 3 + 4i is a root, there should also be a root at 3 - 4i. Therefore, with these four complex roots and the additional single real root of 1, the polynomial is, at least, of degree 5.

A fundamental theorem of algebra states that a polynomial of degree n has exactly n complex roots, counting multiplicity. Hence, the degree of the polynomial is the highest power of the variable x in the polynomial, which is equal to the number of roots if all roots are distinct. The polynomial in question must be at least degree 5 because there must be at least 5 roots including the complex conjugates and the single real root given.