📚 Topic Summary
Writing polynomial functions from zeros involves using the given zeros to construct the factors of the polynomial. Each zero, $x = a$, corresponds to a factor $(x - a)$. If a zero has a multiplicity, it means the factor appears multiple times. By multiplying these factors together, you obtain the polynomial function. Remember to consider a leading coefficient 'a' to account for vertical stretches or compressions. Sometimes, you'll have complex roots which always come in conjugate pairs!
🧮 Part A: Vocabulary
Match the following terms with their definitions:
| Term | Definition |
|---|
| 1. Zero of a function | A. The number of times a root appears in a polynomial. |
| 2. Factor | B. A value of x that makes the function equal to zero. |
| 3. Multiplicity | C. A polynomial with two terms. |
| 4. Binomial | D. Numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit. |
| 5. Complex Numbers | E. An expression that divides evenly into a polynomial. |
✍️ Part B: Fill in the Blanks
To write a polynomial function from its zeros, you first write the function in factored form. Each zero 'r' becomes a factor (x - r). If a zero has a multiplicity of 'n', then the factor (x - r) is raised to the power of ______. Remember that complex zeros always come in ______ pairs. Finally, ______ the factors to obtain the polynomial in standard form.
🤔 Part C: Critical Thinking
Explain in your own words, the significance of complex conjugate roots when constructing polynomial functions with real coefficients. Give an example.