Question

Use the Intermediate Value Theorem to show that the polynomial f(x) =2x4 - 10x2 + 2 has a real zero between
-2 and 0.

...

Select the correct choice below and fill in the answer boxes to complete your choice.

O A. Because f(x) is a polynomial with f(-2) =
-2 and 0.

O B. Because f(x) is a polynomial with f( -2)=
-2 and 0.

O C. Because f(x) is a polynomial with f(-2) =
-2 and 0.

O D. Because f(x) is a polynomial with f(-2) =
-2 and 0.

> 0 and f(0)=

> 0, the function has a real zero between

< 0 and f(0)=

< 0, the function has a real zero between

> 0 and f(0)=

<0, the function has a real zero between

< 0 and f(0)=

> 0, the function has a real zero between

Answer 1

To apply the Intermediate Value Theorem (IVT) to show that the polynomial \( f(x) = 2x^4 - 10x^2 + 2 \) has a real zero between -2 and 0, we need to evaluate the function at the endpoints of the interval and check the sign of the function values.

1. **Evaluate \( f(-2) \):**
\[
f(-2) = 2(-2)^4 - 10(-2)^2 + 2
\]
\[
= 2(16) - 10(4) + 2
\]
\[
= 32 - 40 + 2 = -6
\]

2. **Evaluate \( f(0) \):**
\[
f(0) = 2(0)^4 - 10(0)^2 + 2
\]
\[
= 0 - 0 + 2 = 2
\]

Now we have:
- \( f(-2) = -6 \) (which is less than 0)
- \( f(0) = 2 \) (which is greater than 0)

Since \( f(-2) < 0 \) and \( f(0) > 0 \), by the Intermediate Value Theorem, there exists at least one real zero of \( f(x) \) in the interval \( (-2, 0) \).

Using the provided options, the correct choice would be:

**O B. Because f(x) is a polynomial with \( f(-2) = -6 < 0 \) and \( f(0) = 2 > 0 \), the function has a real zero between -2 and 0.**

Fill in:
- \( f(-2) = -6 \) and \( f(0) = 2 \)
- the function has a real zero between -2 and 0.