Let's work through each part of the problem.

We are asked to form polynomial functions based on the given zeros, their multiplicities, and the degrees. We'll assume the stretch-compression factor is 1, meaning the leading coefficient is 1.

a) Zeros: -5, -2, 4; Degree: 3

• The polynomial will have factors of $(x + 5)$, $(x + 2)$, and $(x - 4)$ since the zeros are $-5$, $-2$, and $4$.
• The polynomial is: $P(x) = (x + 5)(x + 2)(x - 4)$ We expand this: $P(x) = (x + 5)(x + 2)(x - 4)$ First, expand $(x + 5)(x + 2)$: $(x + 5)(x + 2) = x^2 + 7x + 10$ Now multiply by $(x - 4)$: $P(x) = (x^2 + 7x + 10)(x - 4)$ $P(x) = x^3 - 4x^2 + 7x^2 - 28x + 10x - 40$ $P(x) = x^3 + 3x^2 - 18x - 40$

b) Zeros: 7, multiplicity 1; -4, multiplicity 3; -3, multiplicity 1; Degree: 5

• The factors are $(x - 7)$, $(x + 4)^3$, and $(x + 3)$.
• The polynomial is: $P(x) = (x - 7)(x + 4)^3(x + 3)$ We expand step by step. First, expand $(x + 4)^3$: $(x + 4)^3 = (x + 4)(x + 4)(x + 4)$ Expanding the first two factors: $(x + 4)(x + 4) = x^2 + 8x + 16$ Now, multiply by $(x + 4)$: $(x^2 + 8x + 16)(x + 4) = x^3 + 12x^2 + 48x + 64$ Now multiply by $(x - 7)$: $(x - 7)(x^3 + 12x^2 + 48x + 64)$ $P(x) = x^4 + 5x^3 - 36x^2 - 288x - 448$ Finally, multiply by $(x + 3)$: $(x^4 + 5x^3 - 36x^2 - 288x - 448)(x + 3)$ Expanding: $P(x) = x^5 + 8x^4 - 21x^3 - 396x^2 - 1312x - 1344$

c) Zeros: 3, multiplicity 2; 0, multiplicity 2; -1, multiplicity 1; Degree: 5

• The factors are $(x - 3)^2$, $x^2$, and $(x + 1)$.
• The polynomial is: $P(x) = (x - 3)^2x^2(x + 1)$ First, expand $(x - 3)^2$: $(x - 3)^2 = x^2 - 6x + 9$ Now, multiply by $x^2$: $(x^2 - 6x + 9)x^2 = x^4 - 6x^3 + 9x^2$ Now multiply by $(x + 1)$: $(x^4 - 6x^3 + 9x^2)(x + 1)$ Expanding: $P(x) = x^5 - 5x^4 + 3x^3 + 9x^2$

Would you like a further breakdown of any of these steps?

Here are five related questions for practice:

1. How do you handle multiplicities when forming polynomials from zeros?
2. What happens to a polynomial's degree when one of the zeros has a multiplicity greater than 1?
3. How do you expand polynomials with multiple terms like $(x + 4)^3$?
4. What role does the leading coefficient play in the shape of the polynomial?
5. How does the Fundamental Theorem of Algebra apply to the number of real zeros a polynomial can have?

Tip: Always double-check your factorizations and expansions to avoid mistakes in sign or arithmetic, especially when dealing with higher-degree polynomials.