1. **State the problem:** Create a polynomial function in factored form with three real zeros based on the birthday 05/27/07. 2. **Identify the zeros:** The zeros are 5 (month), 27 (day), and 7 (year last two digits). So zeros are $x=5$, $x=27$, and $x=7$. 3. **Multiplicity conditions:** - At least one zero with even multiplicity. - At least one zero with odd multiplicity. 4. **Degree condition:** The degree equals the number of letters in the last name. Since the last name is not given, assume a last name with 6 letters (example: "Smithy"). So degree = 6. 5. **Assign multiplicities:** We have 3 zeros and degree 6, so multiplicities must sum to 6. - Let zero 5 have multiplicity 2 (even). - Let zero 27 have multiplicity 3 (odd). - Let zero 7 have multiplicity 1 (odd). 6. **Vertical compression:** Multiply the polynomial by a factor $0 < a < 1$, for example $a=\frac{1}{2}$. 7. **Write the polynomial:** $$ P(x) = \frac{1}{2}(x-5)^2 (x-27)^3 (x-7)^1 $$ 8. **Summary:** The polynomial has zeros 5, 27, 7 with multiplicities 2, 3, 1 respectively, degree 6, and vertical compression by $\frac{1}{2}$. This satisfies all criteria.