Final answer:
We find the zeros of the cubic polynomial f(x) = x³ – 9x² + 28x – 30 using the Linear Factors Theorem. The first zero is found by testing factors of the constant term, leading to the discovery that x = 1 is a zero. Division of the polynomial by (x - 1) yields a quadratic equation that can be solved to find the remaining zeros.
Step-by-step explanation:
The question involves finding all zeros of a cubic polynomial f(x) = x³ – 9x² + 28x – 30 using the Linear Factors Theorem. To solve this, we look for a factor of the constant term, 30, that when plugged into f(x) yields zero. By testing factors such as ± 1, ± 2, ± 3, and so on, we find that f(1) = 0, which tells us that x = 1 is a zero of the polynomial.
After finding this zero, we can perform polynomial division or synthetic division to divide f(x) by x-1. This will result in a quadratic equation which can then be solved using the quadratic formula. The general form of the quadratic equation is ax² + bx + c = 0.
After performing the division, we get a quadratic polynomial, which can then be factored or solved to find the remaining zeros of the original cubic polynomial. If any zero is repeated, it is said to have multiplicity greater than one. The Linear Factors Theorem dictates that a polynomial of degree n will have exactly n roots, including multiplicities.