Download Les. 6.7 Roots and Zeros.notebook

Les. 6.7 Roots and Zeros.notebook
January 28, 2015
Section 6.7
Roots and Zeros
Objectives:
1. To determine the number of zeros and maximum number of turning points of a polynomial function.
2. To find the zeros of a polynomial function.
Jan 19­5:34 PM
Fundamental Theorem of Algebra
Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.
There was a lot of hoopla when it was proven; there is only one reason it is important­its corollary.
Corollary to the Fundamental Theorem of Algebra
A polynomial equation of degree n has exactly n roots in the set of complex numbers, including repeated roots.
State the number of roots for each function:
x6 ­ 5x4 + 3x2 ­ 6x + 7
7x2 ­ 7x + 9
Jan 19­5:35 PM
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Les. 6.7 Roots and Zeros.notebook
January 28, 2015
Turning point
A point where the graph turns or changes direction
Turning points correspond to the relative maximums and relative minimums. Number of Turning Points of Polynomial Functions The graph of a polynomial function of degree n has at most n – 1 turning points.
If a polynomial function has n distinct zeros, then its graph will have exactly n – 1 turning points.
Jan 20­2:56 PM
For each polynomial, state the number of zeros and the maximum number of turning points:
3x5 ­ 7x4 ­ 5x3 + 6x ­ 4
7x8 ­ 4x6 + 2x4 + 5x3 ­ 8x2 + 16x ­ 44
5x3 ­ x + 9
Jan 20­6:06 PM
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Les. 6.7 Roots and Zeros.notebook
January 28, 2015
Complex Conjugates Theorem
If a + bi is a zero of a polynomial function with real coefficients, then a ­ bi is also a real zero of the function;
provided a and b are real numbers such that b ≠0.
This means that complex zeros containing an imaginary part will ALWAYS come in pairs.
Given 7 ­ 4i is a zero, state another zero of the function:
Jan 19­5:35 PM
Steps to write a polynomial function of least degree with integral coefficients that has the given zeros:
1. Find any additional (complex) zeros
2. Convert the zeros to factors
3. Write the function as a product of all of its factors
4. Expand the polynomial by multiplying out all of the
factors.
Jan 19­5:50 PM
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Les. 6.7 Roots and Zeros.notebook
January 28, 2015
Write a polynomial function of least degree with integral coefficients that has the given zeros:
4, 3, ­2i Jan 19­5:50 PM
Write a polynomial function of least degree with integral coefficients that has the given zeros:
7, 2 ­ 5i Jan 19­5:50 PM
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Les. 6.7 Roots and Zeros.notebook
January 28, 2015
Write a polynomial function of least degree with integral coefficients that has the given zeros:
1, 1, 2, 0
Jan 19­5:50 PM
Assignment:
p 388: 17-26 all, 27-31 odd, 43-47 odd
Jan 19­5:56 PM
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