Polynomial Functions and Zeros

A polynomial function of degree $n$ has exactly $n$ complex zeros, counting multiplicities.
Key Terms:
- Degree: The highest power of the variable in the polynomial.
- Complex Zeros: Includes both real and imaginary numbers.
- Multiplicity: The number of times a particular zero occurs (e.g., if $x-a$ is a factor repeated $k$ times, $a$ is a zero with multiplicity $k$).

A polynomial function always has $n$ complex zeros. This means that both real zeros and imaginary zeros are counted towards the total zeros.
Example: A degree five polynomial can have various combinations of real or imaginary zeros, including repeated zeros.
- E.g., For a polynomial $P(x)$ of degree 5:
- Possible zeros: $1, 1, -rac{1}{2}, i ext{sqrt}{3}, -i ext{sqrt}{3}$ (with $1$ being counted twice).

Complex Number Definition: Any number in the form $a + bi$, where:
- $a$ is the real part,
- $b$ is the imaginary part.
Real numbers can be seen as complex numbers with the imaginary part set to zero (e.g., $6 + 0i$).
Imaginary Numbers: Numbers in the form $bi$, where $b$ is a real number. Examples include:
- Purely Imaginary: $3i$, $-8i$, and $ ext{sqrt}{3}i$.
- Complex Imaginary: Numbers like $6 + 3i$, $7 - 2i$.
Complex numbers include both real and imaginary numbers.

If asked to find complex zeros, you must identify both real and imaginary solutions.
If given a polynomial that is degree $n$, it will have exactly $n$ complex zeros, regardless of their nature (real or imaginary).

Imaginary zeros come in conjugate pairs. This means if $a + bi$ is a zero of a polynomial, then $a - bi$ must also be a zero.
- Example: If $3 + 7i$ is a zero, then $3 - 7i$ is also a zero.
Imaginary zeros must come from solving situations where square roots yield imaginary solutions, hence they appear in pairs due to the symmetry in their formation through square roots.

For a polynomial stated as follows: P(x) = (x - 1)(x - 1)(x - (-rac{1}{2}))(x - (i ext{sqrt}{3}))(x - (-i ext{sqrt}{3}))
- Here, you see that $(x - 1)$ appears twice, indicating multiplicity of 2.
- Total: 5 zeros counted as $1, 1, -rac{1}{2}, i ext{sqrt}{3}, -i ext{sqrt}{3}$ indicating both real and imaginary roots are accounted for.

All polynomials that can be factored meet specific criteria, such as:
- Always factoring into $n$ linear factors for a degree $n$ polynomial.
- This includes all chapter two polynomials within certain bounds as long as they are solvable within real or complex domains.

To find the multiplicity of a root, perform synthetic division to see if the root produces a remainder of zero consecutively.
- Use this to determine further roots until none can be found, indicating either single or multiple zeros.
Example of finding multiplicity $2$ by checking if synthetic division yields $0$ again after already finding it zero once.

These can either be graphical or analytical and often require sketching or testing intervals based on zeros.
Inequalities like $P(x) < 0$ or $P(x) > 0$ prompt examining intervals between the zeros to determine where the polynomial is on the correct side (above or below the x-axis).
Example Logically: If $f(x)$ is a sketch of polynomial, determine where the sketch lies inside bounds of x-axis:
- Where $P(x) < 0$ (below x-axis) or $P(x) > 0$ (above x-axis).

Critical Points: Found from zeros of the polynomial; use them to split the number line.
Choose sample points from each interval to test the inequality:
- If $f(x)=P(x)$ yields positive, the polynomial is above the x-axis in that interval, else it’s negative.

The polynomial $ ext{x}^2 - 13x + 42 = 0$ factors to $(x - 6)(x - 7)$.
Analyzing graphical positions allows to conclude where $f(x) < 0$ provides the solutions directly from the zeroes.
The interval notation for these solutions is denoted as $(- ext{infinity}, -6) igcup (-6, -7)$ for appropriate signs from $<0$.

Always ensure to clarify use of brackets ([ ]) for inclusive treatment of equalities in inequalities versus parentheses (or ) for strictly greater or lesser functions.
Graphical representation and sign analysis provides the easiest way to derive meaningful solutions while still adhering to polynomial rules and structures.