Question

Which of the values shown are potential roots of f(x)=3x^3-13x^2-3x+45 ? Select all that apply.

a)± 1/3
b)± 5
c)± 5/3
d)± 9
e)± 1
f)± 15
g)± 3
h)± 45

Answer 1

The potential roots of the polynomial f(x)=3x^3-13x^2-3x+45 according to the Rational Root Theorem are ± 1 and ± 3, which correspond to options e) ± 1 and g) ± 3 from the list provided.

Step-by-step explanation:

The correct answer is option e) ± 1 and g) ± 3. To find the potential roots of a polynomial, we can use the Rational Root Theorem, which states that for a polynomial equation with integer coefficients, any rational solution, written in its lowest terms p/q, p is a factor of the constant term and q is a factor of the leading coefficient. In the case of the polynomial f(x)=3x^3-13x^2-3x+45, the constant term is 45 and the leading coefficient is 3.

By listing the factors of 45 (±1, ±3, ±5, ±9, ±15, ±45) and the factors of 3 (±1, ±3), and forming potential rational roots by dividing the factors of the constant term by the factors of the leading coefficient, we find the possible rational roots to be ±1, ±5, ±1/3, and ±5/3, as well as their negatives. Thus, the correct options from the provided list are e) ± 1 and g) ± 3, as these are included in the list of potential rational roots according to the theorem.