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To find a degree 4 polynomial with the given zeros and the leading coefficient of \( x^4 \) equal to 1, we can use the following approach.

### Given zeros:
- \( -7 \)
- \( -1 \)
- \( 1 \)
- \( 5 \)

### General form of a polynomial:
If \( a \), \( b \), \( c \), and \( d \) are the roots of a polynomial, then the polynomial can be written as:
\[
P(x) = (x - a)(x - b)(x - c)(x - d)
\]

Here, the zeros are \( -7 \), \( -1 \), \( 1 \), and \( 5 \). Thus, the polynomial is:
\[
P(x) = (x + 7)(x + 1)(x - 1)(x - 5)
\]

### Expanding the polynomial:
1. First, group and expand two pairs of factors:
\[
(x + 7)(x + 1) = x^2 + 8x + 7
\]
\[
(x - 1)(x - 5) = x^2 - 6x + 5
\]

2. Now, multiply these two quadratic expressions:
\[
(x^2 + 8x + 7)(x^2 - 6x + 5)
\]

Using the distributive property (also known as FOIL for binomials), we expand this:
\[
= x^2(x^2 - 6x + 5) + 8x(x^2 - 6x + 5) + 7(x^2 - 6x + 5)
\]
\[
= x^4 - 6x^3 + 5x^2 + 8x^3 - 48x^2 + 40x + 7x^2 - 42x + 35
\]

3. Combine like terms:
\[
= x^4 + (-6x^3 + 8x^3) + (5x^2 - 48x^2 + 7x^2) + (40x - 42x) + 35
\]
\[
= x^4 + 2x^3 - 36x^2 - 2x + 35
\]

So, the degree 4 polynomial is:
\[
P(x) = x^4 + 2x^3 - 36x^2 - 2x + 35
\]

Would you like further details on how this expansion works or any clarifications?

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Here are 5 related questions you might find interesting:
1. How do you find the degree of a polynomial given its factored form?
2. How do the signs of the zeros affect the polynomial expression?
3. What happens if the coefficient of \(x^4\) is not 1? How do you scale the polynomial?
4. How can the quadratic terms be expanded systematically?
5. How would the process change if there were complex roots?

**Tip:** To verify if the expanded polynomial is correct, you can substitute each root back into the polynomial to see if it equals zero.

Find a degree 4 polynomial having zeros -7, -1, 1, and 5 and the coefficient of x^4 equal to 1.

Learn how to find a degree 4 polynomial with specific zeros: -7, -1, 1, and 5, with the leading coefficient of x^4 set to 1. This solution includes a detailed algebraic expansion and step-by-step instructions on polynomial multiplication. Ideal for students in Grades 9-12.

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