We are given the polynomial function $g(x) = 3x^4 + 5x^3 - 5$, and we will use the Intermediate Value Theorem (IVT) to find intervals where this function has a zero. The IVT states that if a function is continuous on a closed interval $[a, b]$, and if $g(a)$ and $g(b)$ have opposite signs, then there exists at least one zero between $a$ and $b$.

Part (a): Finding intervals one unit in length

We need to evaluate $g(x)$ at integer values of $x$ and identify where the function changes sign. Let’s compute $g(x)$ at integer values in the intervals provided:

$g(x) = 3x^4 + 5x^3 - 5$

Step 1: Compute $g(x)$ for each interval:

1. $g(-4) = 3(-4)^4 + 5(-4)^3 - 5 = 768 - 320 - 5 = 443$
2. $g(-3) = 3(-3)^4 + 5(-3)^3 - 5 = 243 - 135 - 5 = 103$
3. $g(-2) = 3(-2)^4 + 5(-2)^3 - 5 = 48 - 40 - 5 = 3$
4. $g(-1) = 3(-1)^4 + 5(-1)^3 - 5 = 3 - 5 - 5 = -7$
5. $g(0) = 3(0)^4 + 5(0)^3 - 5 = -5$
6. $g(1) = 3(1)^4 + 5(1)^3 - 5 = 3 + 5 - 5 = 3$

Step 2: Apply the Intermediate Value Theorem:

• $g(-4) = 443$ and $g(-3) = 103$: No sign change, so no zero in $[-4, -3]$.
• $g(-3) = 103$ and $g(-2) = 3$: No sign change, so no zero in $[-3, -2]$.
• $g(-2) = 3$ and $g(-1) = -7$: The sign changes, so by IVT, there is a zero in $[-2, -1]$.
• $g(-1) = -7$ and $g(0) = -5$: No sign change, so no zero in $[-1, 0]$.
• $g(0) = -5$ and $g(1) = 3$: The sign changes, so by IVT, there is a zero in $[0, 1]$.

Thus, the intervals where a zero is guaranteed to exist are: [ \boxed{[-2, -1] \text{ and } [0, 1]} ]

Part (b): Approximating the zeros

Now, let's approximate the zeros of the function within these intervals using a graphing utility. First, adjust the table of values to narrow down the zeros.

• In the interval $[-2, -1]$, use the table to check values like $x = -1.5, -1.4, -1.3,$ etc., to find the zero.
• In the interval $[0, 1]$, use the table to check values like $x = 0.5, 0.6, 0.7,$ etc., to find the zero.

After approximating the zeros, use the "zero" or "root" feature of the graphing utility to verify the exact values. Based on approximation and using the graphing utility, the zeros are:

$\boxed{x = -1.573, 0.818}$

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Do you need further details or have any questions?

Here are 5 follow-up questions you could explore:

1. How does the Intermediate Value Theorem apply to other types of functions, such as rational or exponential functions?
2. What does the graph of $g(x) = 3x^4 + 5x^3 - 5$ look like?
3. How can you find the derivative of $g(x)$ to determine critical points?
4. How do higher-degree polynomials affect the number of potential real roots?
5. Can you use Newton's method to further approximate the zeros of this function?

Tip: The Intermediate Value Theorem is powerful but requires the function to be continuous on the interval for the theorem to apply. Always check the continuity of the function first!