Download Section 8.4: Roots and Zeros

Section 8.4: Roots and Zeros
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Fundamental Theorem of Algebra:
Every polynomial equation with degree greater than zero has at
least one root in the set of complex numbers (real or imaginary)
Corollary:
A polynomial equation with degree n and with complex
coefficients has exactly n roots in the set of complex numbers
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Examples:
1. Find all roots of 0 = x3 + x2 - 4x - 4
f(x) = 0 means x - intercepts
Strategies:
1. graph it to look for real zeros
-table
-trace
-2nd calc (zeros)
2. Use synthetic division to get to "x2" term
3. Solve the quadratic (factor, QF)
Complex Conjugate Theorem:
Suppose a and b are real numbers with b ≠ 0. If a + bi
is a zero of a polynomial function, then a - bi is also a
zero of the function.
2.
Given a function and one of its zeros, find the remaining zeros
of the function.
a.
f(x) = x3 - 7x2 + 16x - 10; 3 - i
graph on calculator
Strategies:
1. graph it to look for real zeros
-table
-trace
-2nd calc (zeros)
2. Use synthetic division to get to "x2" term
3. Solve the quadratic (factor, QF)
b.
g(x) = x3 - 2x2 + 5x + 26; -2
Descartes' Rule of Signs:
If we have a polynomial in descending order then:
- the number of positive real zeros of the polynomial is the same as the number of
changes in sign of the coefficient of the terms, or less than this by an even number,
and
- the number of negative real zeros of the polynomial is the same as the number of
changes in sign of the coefficient of the terms of p(-x), or less than this by an even
number.
3.
State the number of positive real zeros, negative real zeros,
and imaginary zeros for
a.
b.
g(x) = -9x3 - x2 + 5x - 8
g(x) = x5 - x4 + x3 + x - 7
4.
Write the polynomial function of least degree with integer
coefficients whose zeros include the following.
a. 1 and 1 + i
b. -2 and 2 + 3i