Given f(x): 2x^4 + ax^3 - 6x^2 + 10x - 84, and knowing 3 is a root of f(x), which is the value of a?

For this exercise we need to be aware of the factor theorem.The factor theorem states when f(c) = 0 then x - c is a factor of the polynomial f(x) and c is a root of f(x). Because we are told that one of the roots of f(x) is 3, we know than when replacing x for 3 the result of our polynomial will be 0.We then need to solve the equation f(3) = 0.f(3) = 2 * 3⁴ + a * x 3³ - 6 * 3² + 10 * 3 - 84 = 0162 + 27a - 54 + 30 - 84 = 027a = 84 - 30 + 54 - 16227a = -54a = -2We have arrived at a value for a. We can test that by running again f(3) this time with a = -2f(3) = 2 * 3⁴ - 2 * x 3³ - 6 * 3² + 10 * 3 - 84162 - 54 - 54 + 30 - 84 = 192 - 192 = 0Now that we solved the equation with our solution, we can confidently say that a = -2.