Final answer:
To find the factors of the given polynomial, we can use synthetic division or long division to test the possible rational roots. The factors of the polynomial are (x+4)(x+3)(2x-3).
Step-by-step explanation:
To find the factors of the given polynomial, we can use the synthetic division method. First, we need to find the possible rational roots of the polynomial. The rational roots theorem states that the possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is -72 and the leading coefficient is 2, so the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±36, ±72.
Using synthetic division or long division, we can test these possible roots to see if they are actual roots of the polynomial. By dividing the polynomial by a possible root, we can check if the remainder is zero. If the remainder is zero, then the possible root is a factor of the polynomial.
If we perform synthetic division with the possible roots, we find that the factors of the polynomial are (x+4), (x+3), and (2x-3). Therefore, the factors of the polynomial 2x³ - 23x² - 78x - 72 are (x+4)(x+3)(2x-3).