Fundamental Theorem Of Algebra Quiz

Explanation
The equation is a quadratic equation in the form of x^2 + 9 = 0. To solve this equation, we can subtract 9 from both sides, resulting in x^2 = -9. Since the square of any real number cannot be negative, we conclude that there are no real solutions for x. However, we can introduce the concept of imaginary numbers, denoted by "i", where i^2 = -1. Thus, the solutions to the equation are x = ±3i.

Explanation
The given polynomial f(x) is a fourth-degree polynomial. In order to find its zeros, we need to set f(x) equal to zero and solve for x. The zeros of a polynomial are the values of x for which the polynomial evaluates to zero.

By factoring the polynomial, we can rewrite it as (x - 3)(x + 7)(x - 2i)(x + 2i). This means that the zeros of the polynomial are x = 3, x = -7, x = 2i, and x = -2i.

However, since the question asks for a linear factorization, we can rewrite the complex zeros in their conjugate pairs: x = 3, x = -7, and x = ±2i.

Therefore, the correct answer is x = 3, -7, ±2i.

Explanation
The given answer, (X-2)^3 (X+4)^2, is the correct polynomial of minimum degree in factored form with real coefficients. It satisfies the given conditions of having zeros at 2 with multiplicity 3 and -4 with multiplicity 2. The factors (X-2)^3 and (X+4)^2 represent the zeros and their respective multiplicities. Therefore, the answer is (X-2)^3 (X+4)^2.

Explanation
The given function is a cubic function, which means it has a degree of 3. According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n complex zeros. In this case, the function has a degree of 3, so it will have 3 complex zeros. However, the question specifically asks for the number of real and imaginary zeros. A complex zero consists of both a real and an imaginary part. Since the function has 3 complex zeros, it means there are 3 imaginary zeros and 1 real zero. Therefore, the correct answer is Real: 1, Imaginary: 2.

Explanation
The given function is a cubic function, which means it has three possible zeros. However, the question asks for the number of real and imaginary zeros. In this case, the function has one real zero and two imaginary zeros.