Overview
This lecture introduces complex zeros in polynomials, reviews key factoring and graphing concepts, explains multiplicity and sign analysis, covers the nature and pairing of real and non-real (complex) zeros, and presents a method to deduce a polynomial's degree from output tables.
Zeros and Roots of Polynomials
- A zero (or root) of a function is an input value where the output equals zero.
- If the zero is real, it corresponds to an x-intercept at (a, 0).
- The factor associated with a real zero a is (x - a).
Factoring to Find Zeros
- Factor polynomials by pulling out greatest common factors and factoring trinomials.
- Set each factor equal to zero (zero product property) to find x-intercepts.
- Example: For x³ - 2x² - 8, zeros are x = 0, x = 4, and x = -2.
Sign Analysis and Intervals
- For inequalities, factor the polynomial and find zeros.
- Divide the number line into intervals based on zeros and test the sign of the function in each interval.
- Use interval notation to specify where the function is positive, negative, or zero.
Multiplicity of Roots
- Multiplicity of a root is the exponent n of its linear factor (e.g., (x - a)ⁿ).
- Even multiplicity: graph bounces at x=a; sign does not change.
- Odd multiplicity: graph crosses at x=a; sign changes.
Graphing Using Multiplicity and Signs
- Build a sign table for intervals between zeros.
- Determine sign changes at each zero based on multiplicity (bounce or cross).
- Use one test point in each interval to deduce signs for the whole interval.
Complex and Non-Real Zeros
- Polynomials of degree n have exactly n complex zeros (real and non-real combined, counting multiplicities).
- Non-real zeros always occur in conjugate pairs: if a+bi is a zero, so is a–bi.
- The number of non-real zeros is always even or zero.
Real vs Non-Real Zeros and Degree
- The sum of real and non-real zeros (with multiplicities) equals the polynomial degree.
- Use graph analysis or given zeros (with multiplicities) to deduce the number of non-real zeros.
Successive Differences and Degree from Tables
- For a table with equal input intervals, compute differences between outputs:
- First differences: difference between successive outputs.
- Repeat differences (second, third, etc.) until differences become constant.
- The degree of the polynomial equals the number of times differences are taken to get a constant.
Key Terms & Definitions
- Zero/Root — Value where function output is zero (f(a) = 0).
- Multiplicity — Number of times a particular zero is repeated (exponent of its factor).
- x-intercept — Point where the graph crosses the x-axis (output is zero).
- Complex Zero — Zero that may be real or non-real (involving imaginary numbers).
- Conjugate Pair — Two zeros of the form a+bi and a–bi.
- Successive Differences — Sequence of differences used to find the polynomial degree from a table.
Action Items / Next Steps
- Practice factoring polynomials and identifying zeros.
- Use sign tables and interval notation for polynomial inequalities.
- Complete homework on finding degrees using successive differences and classifying zeros.