Question

What are the zeros of the polynomial function?

f(x)=x^4−2x^3−103x^2+200x+300

Select each correct answer.

1.) −10
2.) −3
3.) −1
4.) 0
5.) 3
6.) 10

Answer 1

To find the zeros of the polynomial function f(x) = x^4 - 2x^3 - 103x^2 + 200x + 300, we need to solve for x when f(x) equals zero.

Using factoring or synthetic division, we can determine the zeros of the polynomial:

By synthetic division, we can divide the polynomial by (x + 10):

-10 | 1 -2 -103 200 300

| -10 120 -800

-----------------------

1 -12 17 -600 -500

The result of the division is x^3 - 12x^2 + 17x - 600.

By synthetic division again, we can divide the remaining polynomial by (x - 3):

3 | 1 -12 17 -600

| 3 -27 -30

------------------

1 -9 -10 -630

The result of the division is x^2 - 9x - 10.

Now, we can factor the quadratic x^2 - 9x - 10:

(x^2 - 9x - 10) = (x - 10)(x + 1)

Therefore, the factored form of the polynomial is:

f(x) = (x + 10)(x - 3)(x - 10)(x + 1)

The zeros of the polynomial function f(x) = x^4 - 2x^3 - 103x^2 + 200x + 300 are:

1.) -10

2.) -3

3.) 10

4.) -1

So, the correct answers are 1.), 2.), 3.), and 4.).