Question

Find an expression for a third-degree polynomial with three distinct zeros in factored form. Why does a polynomial with three distinct zeros exhibit this particular factorization?

a) (f(x) = (x - a)(x - b)(x - c)), where (a), (b), and (c) are distinct zeros.
b) (f(x) = (x - a)^3), where (a) is a repeated zero.
c) (f(x) = (x - a)(x - b)^2), where (a) and (b) are distinct zeros.
d) (f(x) = (x - a)^2(x - b)), where (a) and (b) are repeated zeros.

Answer 1

A third-degree polynomial with three distinct zeros is factored as (f(x) = (x - a)(x - b)(x - c)), where a, b, and c are the distinct zeros, due to the Fundamental Theorem of Algebra.

Step-by-step explanation:

The expression for a third-degree polynomial with three distinct zeros in factored form is (f(x) = (x - a)(x - b)(x - c)), where a, b, and c are the distinct zeros of the polynomial. The reason a polynomial with three distinct zeros exhibits this particular factorization is due to the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots (including multiplicity). Since this polynomial is third-degree, it should have three roots. If all zeros are distinct, each zero corresponds to a linear factor of the form (x - zero). Thus, for three distinct zeros, the polynomial can be factored into three linear factors, each representing a zero of the polynomial.