question of math

Question

What are the roots of f(x) = x4 – 16?

1. 2,2,2i,2i
2. –2, 2, –2i, 2i
3. –2,–2,2,2
4. 2i, 2i, –2i, –2i

Answer 1

Hello Clarissa! To answer this question, you first need to recognize that x⁴ - 16 is a difference of squares, so you can factor the function to be f(x) = (x² - 4)(x² + 4). Since x² - 4 is also a difference of squares, you can then factor this function further to be f(x) = (x + 2)(x - 2)(x² +4).

Finally, you can solve for the roots (also known as the "zeros" or "x-intercepts") by setting each factor equal to zero (to find the value of x when y is zero). So we can say that x + 2 = 0, x - 2 = 0, and x² + 4 = 0. When we solve each of these equations, we find that x = -2, x = 2, and x = ± 2i (since solving x² + 4 = 0 leads to us finding the square root of -4 which gives us the two imaginary solutions of ± 2i). Therefore your answer is the second option.

Answer 2

x⁴ - 16

(x² - 4)(x² + 4)

(x + 2)(x - 2)(x + 2i)(x - 2i)