Final answer:
The zeros of the polynomial f(x) = 2x³ - 8x² - 14x + 20 are found by first verifying that x = 1 is a zero using the Factor Theorem, and then performing polynomial division to get a quadratic equation. This quadratic can be solved either by factoring or using the quadratic formula to find the remaining zeros.
Step-by-step explanation:
To find the zeros of the polynomial f(x) = 2x³ - 8x² - 14x + 20 using the Factor Theorem, we will first use the given information that (x - 1) is a factor of the polynomial. Since (x - 1) is a factor, by the Factor Theorem we know that f(1) = 0. We can perform polynomial division or synthetic division to divide the polynomial by (x - 1) and determine the other factors.
After dividing, we get a quotient of 2x² - 6x - 20, which is a quadratic equation. We can then solve the quadratic equation either by factoring, if possible or by using the quadratic formula to find the other zeros of the polynomial. The quadratic formula is given by x = [-b ± √(b² - 4ac)]/(2a), where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
If the quadratic polynomial can be factored, we can find the roots by setting the factors equal to zero. If it cannot be factored easily, we insert a = 2, b = -6, and c = -20 into the quadratic formula to find the remaining zeros of the original polynomial.