📚 Topic Summary
Finding all the zeros of polynomial functions involves determining the values of $x$ for which the polynomial $P(x)$ equals zero. These zeros can be real or complex numbers. Factoring, the Rational Root Theorem, synthetic division, and the quadratic formula are powerful tools for finding these zeros. Understanding the relationship between zeros and factors is crucial: if $r$ is a zero of $P(x)$, then $(x-r)$ is a factor of $P(x)$.
The process often begins with trying to factor the polynomial directly. If factoring is not straightforward, the Rational Root Theorem helps narrow down potential rational zeros. Once a rational zero is found, synthetic division reduces the degree of the polynomial, making it easier to find the remaining zeros. Complex zeros always occur in conjugate pairs, meaning if $a + bi$ is a zero, then $a - bi$ is also a zero.
🧮 Part A: Vocabulary
Match the term to its definition:
| Term |
Definition |
| 1. Zero of a function |
A. A number that, when multiplied by itself, equals a given number. |
| 2. Root |
B. The highest power of the variable in a polynomial. |
| 3. Degree |
C. A value of $x$ that makes the function equal to zero. |
| 4. Factor |
D. Another name for a solution of a polynomial equation. |
| 5. Conjugate Pair |
E. An expression that divides evenly into another expression. |
|
F. Two complex numbers of the form $a + bi$ and $a - bi$. |
✏️ Part B: Fill in the Blanks
The ________ Root Theorem helps identify potential rational zeros of a polynomial. Once a zero is found, ________ division can reduce the degree of the polynomial. Complex zeros always occur in ________ ________.
🤔 Part C: Critical Thinking
Explain how finding the zeros of a polynomial function can help in sketching its graph.