Question

Use the rational zeros theorem to find all the real zeros for f(x) = 13x^4 + 12x^3 - 40x^2 - 36x + 3.

Answer 1

The rational zeros theorem states that if a polynomial has integer coefficients and a rational root, then that root is a quotient of the factors of the constant term divided by the factors of the leading coefficient. In this case, the possible rational roots are ±1/13, ±3/13, ±1/1, and ±3/1. These are all the real zeros of the polynomial.

Step-by-step explanation:

The rational zeros theorem states that if a polynomial has integer coefficients and a rational root r, then r is a quotient of the factors of the constant term divided by the factors of the leading coefficient. In this case, the polynomial is f(x) = 13x^4 + 12x^3 - 40x^2 - 36x + 3.

The factors of the constant term 3 are ±1 and ±3 and the factors of the leading coefficient 13 are ±1 and ±13. Therefore, the possible rational roots are ±1/13, ±3/13, ±1/1, and ±3/1. These are all the real zeros of the polynomial.