Final answer:
To find all the factors of the function f(x) = x³ - 19x + 30, given that f(-5) = 0, we use the Remainder Theorem to deduce (x + 5) is a factor and then use polynomial division or synthetic division to find the quadratic part, which can be factored or solved using the quadratic formula.
Step-by-step explanation:
If f(-5) = 0, then by the Remainder Theorem, we know that x = -5 is a root of the function f(x) = x³ - 19x + 30. This implies that (x + 5) is a factor of f(x). To find all factors of f(x), we can perform polynomial division, dividing f(x) by (x + 5), or we can use synthetic division. After dividing, we get a quadratic equation which can be factored further if it has real roots, or otherwise, we can solve it using the quadratic formula: ax² + bx + c = 0.
Once we have the factored form of the quadratic, we can list all the factors of f(x). If the quadratic doesn't factor neatly, the roots from the quadratic formula will be the remaining factors in the form of (x - root1) and (x - root2). So the factors of f(x) would be (x + 5), (x - root1), and (x - root2), assuming that root1 and root2 are the real roots obtained from solving the quadratic part of f(x).