Question

What are the zeros of the polynomial function f(x)=x⁴-2x³-16x²-2x+15? Select each correct answer.

1) -5
2) negative 5
3) -1
4) negative 1
5) 0
6) 1
7) 3
8) 5

Answer 1

None of the provided answer choices (-5, -1, 0, 1, 3, 5) are zeros of the polynomial function f(x)=x⁴-2x³-16x²-2x+15, as substituting them into the function does not result in a value of zero.

Step-by-step explanation:

The zeros of a polynomial function are the values for which the function equals zero. To find the zeros of the function f(x)=x⁴-2x³-16x²-2x+15, you would typically either factor the polynomial (if it's factorable), or use numerical methods or software to approximate the solutions. However, based on the given choices, we can use the substitution method to check each provided option and see if it yields zero when plugged into the polynomial function.

Let's evaluate each answer choice by plugging them into the polynomial:

• f(-5) = (-5)⁴ - 2(-5)³ - 16(-5)² - 2(-5) + 15 = 625 + 250 - 400 + 10 + 15 = 500
• f(-1) = (-1)⁴ - 2(-1)³ - 16(-1)² - 2(-1) + 15 = 1 + 2 - 16 + 2 + 15 = 4
• f(0) = (0)⁴ - 2(0)³ - 16(0)² - 2(0) + 15 = 15
• f(1) = (1)⁴ - 2(1)³ - 16(1)² - 2(1) + 15 = 1 - 2 - 16 - 2 + 15 = -4
• f(3) = (3)⁴ - 2(3)³ - 16(3)² - 2(3) + 15 = 81 - 54 - 144 - 6 + 15 = -108
• f(5) = (5)⁴ - 2(5)³ - 16(5)² - 2(5) + 15 = 625 - 250 - 400 - 10 + 15 = -20

As none of these computations result in zero, we can conclude that none of the provided options are zeros of the polynomial f(x).