Question

A third degree polynomial with rational coefficients has zeros of x = {1, 3i} and goes through the point (-1, 5). Find the expanded form of this polynomial.

a) x³+2x²−10x+5
b) x³−2x²+10x−5
c) x³+2x²+10x+5
d) x³ - 2x² - 10x - 5

Answer 1

The expanded form of the given polynomial is x³ - 2x² + 10x - 5.

Step-by-step explanation:

The zeros of a polynomial equation are the values of x that make the equation equal to zero. Since the zeros of the polynomial are x = {1, 3i}, we know that the factors of the polynomial are (x - 1) and (x - 3i) (since complex zeros come in conjugate pairs).

Therefore, the polynomial equation can be written as: (x - 1)(x - 3i)(x + 3i). Multiplying these factors together gives us the expanded form of the polynomial: x³ - x²(3i) - x²(-3i) + 3ix² - 3ix + 3ix - 9i².

Simplifying the terms with i and combining like terms gives us the final expanded form of the polynomial: x³ - 2x² + 10x - 5.