Question

Which is the following is the set of real zeros of the function f(x) = (x3 + 1000)(x4 - 160,000)?

A. { -20, -10, 20}

B. { -20, -10, 10, 20}

C. { -10, 20}

D. { -20, 20}

Answer 1

a

Explanation:

Answer 2

A. { -20, -10, 20 }

Explanation:

Given:

The function is given as:

`f(x)=(x^3+1000)(x^4-160000)`

Let us simplify the function.

First, we use the identity
`a^3+b^3=(a+b)(a^2-ab+b^2)`

`x^2+1000= x^3+10^3=(x+10)(x^2-10x+10^2)\\x^3+1000=(x+10)(x^2-10x+100)`

Next, we use the identity
`a^4-b^4=(a-b)(a+b)(a^2+b^2)`

`x^4-160000=x^4-20^4=(x-20)(x+20)(x^2+20^2)=(x-20)(x+20)(x^2+400)`

Now, the function can be rewritten as:

`f(x)=(x+10)(x^2-10x+100)(x-20)(x+20)(x^2+400)`

Now, the zeros are those values of
`x` for which
`f(x)=0`

Now, for
`f(x)=0`, we must have either of the factors 0.

`x+10=0\\x=-10`

`x-20=0\\x=20\\\\x+20=0\\x=-20`

The factors
`x^2-10x+100` and
`x^2+400` can have no zeros as the first one has imaginary roots and second one is always greater than 0 irrespective of the
`x` values.

So, the possible set of zeros are { -20, 10, 20 }.