Solved examples: Writing polynomial functions from zeros with detailed explanations

📚 Quick Study Guide

• 🌱 Zeros and Factors: If $x = a$ is a zero of a polynomial function, then $(x - a)$ is a factor of the polynomial.
• 💡 Building the Polynomial: Given the zeros $a, b, c, ...$, the polynomial function can be written as $f(x) = k(x - a)(x - b)(x - c)...$, where $k$ is a constant.
• 🧮 Complex Zeros: Complex zeros always occur in conjugate pairs. If $a + bi$ is a zero, then $a - bi$ is also a zero.
• 📈 Multiplicity: If a zero $a$ has multiplicity $n$, then the factor $(x - a)$ appears $n$ times in the polynomial.
• ✍️ Standard Form: After constructing the polynomial from its factors, expand and simplify to write it in standard form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$.

Practice Quiz

1. Question 1: A polynomial has zeros at $x = 2$, $x = -3$, and $x = 1$. Which of the following could be the polynomial function?
   1. $f(x) = (x - 2)(x + 3)(x - 1)$
   2. $f(x) = (x + 2)(x - 3)(x + 1)$
   3. $f(x) = (x - 2)(x - 3)(x - 1)$
   4. $f(x) = (x + 2)(x + 3)(x + 1)$
2. Question 2: A polynomial has zeros at $x = -1$ and $x = 4$, with $x = 4$ having a multiplicity of 2. Which function represents this polynomial?
   1. $f(x) = (x - 1)(x + 4)^2$
   2. $f(x) = (x + 1)(x - 4)^2$
   3. $f(x) = (x - 1)(x - 4)^2$
   4. $f(x) = (x + 1)(x + 4)^2$
3. Question 3: A polynomial has a zero at $x = 3i$. What other zero must it have?
   1. $x = -3i$
   2. $x = 3$
   3. $x = -3$
   4. $x = \frac{1}{3i}$
4. Question 4: Write a polynomial function with zeros at $x = 0$, $x = -2$, and $x = 5$.
   1. $f(x) = x(x - 2)(x + 5)$
   2. $f(x) = x(x + 2)(x - 5)$
   3. $f(x) = (x - 0)(x + 2)(x - 5)$
   4. $f(x) = (x - 0)(x - 2)(x + 5)$
5. Question 5: A polynomial has zeros at $x = 1 + i$ and $x = 1 - i$. What is the quadratic factor of this polynomial?
   1. $x^2 + 2x + 2$
   2. $x^2 - 2x - 2$
   3. $x^2 + 2x - 2$
   4. $x^2 - 2x + 2$
6. Question 6: If a polynomial has zeros at $x = -2$ (multiplicity 1), and $x = 3$ (multiplicity 2), which of these is the polynomial?
   1. $f(x) = (x - 2)(x + 3)^2$
   2. $f(x) = (x + 2)(x - 3)^2$
   3. $f(x) = (x - 2)(x - 3)^2$
   4. $f(x) = (x + 2)(x + 3)^2$
7. Question 7: A polynomial has zeros at $x = -1$, $x = 0$, and $x = 4$. If the polynomial passes through the point $(1, -10)$, find the leading coefficient $k$.
   1. $k = 5$
   2. $k = -\frac{5}{2}$
   3. $k = 2$
   4. $k = -2$