2.6--The Fundamental Theorem of Algebra
What you should learn: To use the fundamental theorem of
algebra to determine the number of zeros of polynomial functions,
to find conjugate pairs of complex zeros, and to find all zeros of
polynomials by using calculator and factoring.
Why you should learn it: Finding zeros of polynomial functions
is an important part of solving real-life problems. Finding where
the graph crosses the x-axis has many real-life implications.
For example when something would hit the ground.
When we have a polynomial of degree n we have said that we
can have at most ___ real zeros.
This polynomial has at most ___ real zeros.
EX: Zeros of Polynomial Functions
a. The first degree polynomial
zero:
has exactly one
b. Counting double roots, the seconddegree polynomial function
has exactly two zeros:
c. The third-degree polynomial function
has exactly three zeros:
d. The fourth-degree polynomial function
has exactly four zeros:
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From these examples you can see that the calculator will only
show you the REAL zeros. We will have to use algebraic means
to find the remaining zeros.
EX: Find all zeros of
Use calculator to find real zeros
Use algebraic means to find remaining zeros (synthetic or
polynomial division, factoring, quadratic formula)
Notice that in the last example we got
complex conjugate
zeros. Complex zeros always come in pairs--complex conjugates
EX: Find a 4th degree polynomial function with real coefficients, that has ­1, ­1, and 3i as zeros.
HW: p. 266 1, 3, (9, 13­find by factoring) (17, 25, ­use calc. to find real zeros first) 35, 37, 63, 66 = 10 problems
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