Final answer:
The roots of the polynomial 2x³ + 4x² − 18x − 36 are x = -3, -2, and 3, found using the Rational Root Theorem and the quadratic formula for the resulting quadratic equation after polynomial division.
Step-by-step explanation:
To find all roots of the given polynomial p(x) = 2x³ + 4x² − 18x − 36, we need to identify values of x that satisfy the equation p(x) = 0. One of the methods to solve such cubic equations is to use the Rational Root Theorem to test possible rational roots, which are of the form ±(factors of constant term) / ±(factors of leading coefficient). After finding one rational root, we can perform polynomial division or use synthetic division to reduce the cubic polynomial to a quadratic form, which can then be solved using the quadratic formula −b ± √(b² - 4ac) / 2a.
For this polynomial, we can see that x = -3 is a root since p(-3) = 0. So, we divide the polynomial by x + 3 to obtain the quotient polynomial of degree 2. The resulting quadratic equation can be solved to find the remaining two roots. The correct roots of the given polynomial are x = -3, x = -2, and x = 3, which corresponds to option a).