Pc 2.5 a_notes
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2.5 Zeros of Polynomial FunctionsEssential Questions:1) How do you find the zeros of a polynomial function?
2) What is the rational zero test?
The Fundamental Theorem of Algebra:
If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the ______________________________
Linear Factorization Theorem:
If f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors ______________________________where c1, c2, ... cn are complex numbers.
Ex. 1) Find all zeros and write the linear factorization of the function.
Please note:FTA (fundamental theorem of algebra) and LFT (linear factorization theorem) only tell us that zeros exist, not how to find them.
Methods for finding zeros:
Rational Zero TestIf the polynomial f(x) has __________ coefficients, every rational zero of f has the form:
Where _____ and _____ have no common factors other than 1 and______ = a factor of the ___________________ = a factor of the _____________
Ex. 2) Find the rational zeros of
Ex. 3) Find the rational zeros of
Ex. 4) Find all the real solutions of
What's the difference?Find all rational zeros:
Find all real zeros:
Find all zeros:
Conjugate PairsIf ___________ is a zero of the function, then ___________ is also a zero of the function.
Ex. 5) Find a fourth-degree polynomial with real coefficients that has 3+2i and 4-i as zeros.
Assignment:
2.5A p. 179 #7-19 (odd), 37, 39, 41