Find the equation of a polynomial of degree 4 with leading coefficient of 1 that has zeros at 

xequals=minus−​2, 

minus−​1, 

​2, and 3.5. Consider clicking on the​ "y-intercept" box.

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Part 1

​f(x)equals=enter your response here

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.

Find the equation of a polynomial of degree 4 with leading coefficient of 1 that has zeros at x = -2, -1, 2, and 3.5. Consider clicking on the y-intercept box.

Learn how to find the equation of a degree 4 polynomial with zeros at -2, -1, 2, and 3.5. This algebraic problem involves factoring polynomials and expanding them to find the final equation. Suitable for students in Grades 10-12.

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