Transcript Presentation

Complex Zeros of
Polynomial
By: Mao & Na
Introduction:
What you'll learn
about :
Two Major
Theorems
Complex
Conjugate Zeros
Factoring with
Real Number
Coefficients.
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Video
• http://www.youtube.com/watch?v=J2TYyU
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Two Major Theorems
• Fundamental Theorem of Algebra: A polynomial
function of degree n has n complex zeros( real or nonreal).
Some of these zeros may be repeated.
• Liner Factorization Theorem: If f(x) is a
polynomial function of degree n >0, then f(x) has precisely
n linear factors and
f(x) = a(x –z1)(x –z2) ..... (x -zn)
Where a is the leading coefficient of f(x) and z1, z2,….., zn are
the complex zeros of f(x). The zi are not necessarily
distinct numbers; some may be repeated.
Fundamental Polynomial
Connections in the Complex Case:
The following statements about a polynomial function f are
equivalent if k is a complex number.
1. x = k is a solution (or root) of the equation f(x) = 0
2. is a zero of the function f.
3.x - k is a factor of f(x).
Complex Conjugate Zeros
Suppose that f(x) is a polynomial function with real
coefficients. If a and b are real numbers with b not equal
0 and a + bi is a zero of f(x), then its complex conjugate
a - bi is also a zero of f(x).
Factoring with Real Number
Coefficients
Let f(s) be a polynomial function with real coefficients. The Linear
Factorization Theorem tells us that f(x) can be factored into the form
f(x) = a(x –z1)(x –z2) ..... (x -zn),
Where zi are complex numbers. Recall, however, that nonreal
complex zeros occur in conjugate pairs. The product of x - (a + bi)
and x - (a - bi) is
[x-(a+bi)][x-(a-bi)] = x^(2) - (a-bi)x - (a+bi)x + (a+bi)(a-bi) = x^(2 ) 2ax + ((a^2) + (b^2)).
Factoring with Real Number
Coefficients
Every polynomial function with real coefficients can be written as a
product of linear factors and irreducible quadratic factors, each with real
coefficients.
Polynomial Function of Odd Degree
Every polynomial function of odd degree with real
coefficients has at least one real zero.
Complex Zeros of Polynomial
Find complex zeros of polynomial function :
F(x) = 3x^(4) + 5x^(3) + 25x^(2) + 45x – 18
http://www.youtube.com/watch?v=YoSs6KJ6I9c
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Assessment of Content
Write the polynomial in standard from, and identify the zeros of the function and the x-intercept of it’s graph.
1.f(x)=(x-3i)(x+3i)
A. x2+9; zeros:+-3i: x-intercepts: none
B. x2+6; zeros:+-3i: x-intercepts: 2
2.f(x)=(x-1)(x-1)(x+2i)(x-2i)
A. X4-3x+25x-4x+4
B. X4-2x+5x-8x+4
Write the polynomial function of minimum degree in standard form with real coefficient whose zeros included those
listed.
3. I and – I
A. x3+4
B. x2+1
4. 2,3 and I
A. X4-5x3+7x-5x+6
B. X2-2x2+7x-3x+4
State how many complex and real zeros the function has.
5. f(x)=x2-2x+7
A. 2 complex zeros: none real
B. 4 complex zeros: 7 real
Assessment of Content continue
Find all of the zeros and write a linear factorization of the function.
7. f(x)= x3+4x-5
A. Zeros: x=1, x = -1/2 ± √ 19/2 i ; f(x) = ¼ (x-1)(2x+1+ √19i) (2x+1- √ 19i)
B. Zeros: x=2, x = -1/2 ± √ 19/2 i ; f(x) = ¼ (x-1)(2x+1+ √19i) (2x+1- √ 19i)
8. f(x)= x4+x3+5x2-x-6
A. Zeros: x= ± 1, x = -1/2 ± √ 23/2i ; f(x) = ¼ (x-1)(x+1)(2x+1+ √ 19i) (2x+1- √ 23i
B. . Zeros: x= ± 1, x = -1/2 ± √ 23/2i ; f(x) = ¼ (x-1)(x+1)(2x+1+ √ 19i) (2x+1- √ 23i
Using the given zeros, find all of the zeros and write a linear factorization of f(x).
9. 1+i is a zeros of f(x)=x4-2x3-x2+6x-6
A. Zeros: x=2 x = -1/2 ± √ 19/2 i ; f(x) = ¼ (x-1)(3x+1+ √ 19i) (3x+1- √ 19i)
B. Zeros: x=1, x = -1/2 ± √ 19/2 i ; f(x) = ¼ (x-1)(2x+1+ √ 19i) (2x+1- √ 19i)
10. 3-2i is a zeros of f(x)=x4-6x3+11x2+12x-26
A. Zeros: x +- √ 2. x =3 ± 2i: f(x) = (x- √ 2) ( x+ √ 2)(x-3+2i)(x-3-2i).
B. Zeros: x +- √ 4. x =3 ± 3i: f(x) = (x- √ 2) ( x+ √ 2)(x-3+2i)(x-3-2i).
Answer Key
1.
2.
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5.
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7.
8.
x2+9; zeros:+-3i: x-intercepts: none
X4-2x+5x-8x+4
x2+1
X4-5x3+7x-5x+6
2 complex zeros: none real
4 complex zeros: 2 real
Zeros: x=1, x = -1/2+- √ 19/2 i ; f(x) = ¼ (x-1)(2x+1+ √19i) (2x+1- √ 19i)
Zeros: x= ± 1, x = -1/2+- square root 23/2i ; f(x) = ¼ (x-1)(x+1)(2x+1+square root 19i) (2x+1square root 23i)
9. Zeros: x +- √ 3. x =1 +- i: f(x) = (x- √ 3) ( x+ square 3)(x-1+i)(x-1-i).
10.Zeros: x +- √ 2. x =3 +- 2i: f(x) = (x- √ 2) ( x+ √ 2)(x-3+2i)(x-3-2i).
Works Cited
Demana, Franklin. Waits, Bert K.. Foley, Gregory D.. Kennedy, Daniel. Pre- Calculus. Eight Edition.
Graphical, Numerical, Algebraic.
"Finding Complex Zeros of a Polynomial Function." YouTube. YouTube, 10 Oct. 2011. Web. 21 Jan.
2013.
"X Finds Out His Value." YouTube. YouTube, 31 Dec. 2008. Web. 21 Jan. 2013.