To find the equation of a polynomial of degree 4 with the given conditions, we can use the fact that the zeros of the polynomial are known. The zeros are $x = -2$, $x = -1$, $x = 2$, and $x = 3.5$. The leading coefficient is 1, which means the polynomial can be written as:

$f(x) = a(x + 2)(x + 1)(x - 2)(x - 3.5)$

Since the leading coefficient is 1, we can set $a = 1$, so the polynomial becomes:

$f(x) = (x + 2)(x + 1)(x - 2)(x - 3.5)$

Now let's expand this expression step by step:

1. First, multiply $(x + 2)(x + 1)$: $(x + 2)(x + 1) = x^2 + 3x + 2$

2. Next, multiply $(x - 2)(x - 3.5)$: $(x - 2)(x - 3.5) = x^2 - 5.5x + 7$

3. Finally, multiply these two results together: $(x^2 + 3x + 2)(x^2 - 5.5x + 7)$

Expanding this product will give the final polynomial expression:

$f(x) = x^4 - 2.5x^3 + 0.5x^2 - 26.5x + 14$

So, the polynomial is:

$f(x) = x^4 - 2.5x^3 + 0.5x^2 - 26.5x + 14$

Let me know if you'd like more details on the expansion or any other steps!

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Relative Questions:

1. How do you determine the degree of a polynomial?
2. What is the significance of a leading coefficient in a polynomial?
3. How can you verify that these roots correspond to the given polynomial?
4. Can a polynomial have complex roots, and if so, how does that affect the equation?
5. What happens if you change the value of the leading coefficient in a polynomial?

Tip:

Always factor the polynomial in smaller steps when expanding it to avoid mistakes, especially when dealing with fractional or complex roots.