Question

Which statement about the polynomial function g(x) is true?

-If all rational roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1.
-If all roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1.
-If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers.
-If the leading coefficient of g(x) is 1, all roots of g(x) = 0 must be integers.

Answer 1

The Rational Root Theorem implies that if a polynomial with integer coefficients has a leading coefficient of 1, then all of its rational roots must be integers. However, this does not assure that all roots will be integers as they could be irrational or complex numbers.

Step-by-step explanation:

The correct statement about the polynomial function g(x) is: If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers. This is a derivation from the Rational Root Theorem, which tells us that if a polynomial has integer coefficients and the leading coefficient is 1 (a monic polynomial), then any rational root, expressed as a fraction p/q where p and q are coprime (have no common factors other than 1), must actually have q = 1 because the only possible factors of the leading coefficient (which is 1) are ±1. Therefore, all rational roots must be integers. However, this does not guarantee that all roots will be integers since roots can also be irrational or complex numbers.

Answer 2

C) If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers.

Step-by-step explanation: