every polynomial p(x) of degree n>0 has at least one zero in the complex number system
DESCRIPTION
N Zeros Theorem. Every polynomial P(x) of degree n>0 can be expressed as the product of n linear factors. Hence, P(x) has exactly n zeros, not necessarily distinct. The Fundamental Theorem of Algebra. Every polynomial P(x) of degree n>0 has at least one zero in the complex number system. - PowerPoint PPT PresentationTRANSCRIPT
Every polynomial P(x) of degree n>0 has at least one zero in the complex number system.
N Zeros Theorem
Every polynomial P(x) of degree n>0 can be expressed as the product of n linear factors. Hence, P(x) has exactly n zeros, not necessarily distinct.
The Fundamental Theorem of Algebra
21. ( ) 3 6P x x
Find all zeros. Write the polynomial as the product of linear factors.
21. ( ) 3 6P x x
Find all zeros. Write the polynomial as the product of linear factors.
2
2
0 3 6
2
2
x
x
x i
( ) 2 2P x x i x i
42. ( ) 625P x x
Find all zeros. Factor the polynomial as the product of linear factors.
42. ( ) 625P x x
Find all zeros. Factor the polynomial as the product of linear factors.
4
2 2
0 625
0 25 25
x
x x
( ) 5 5 5 5P x x x x i x i
2 225 0 or 25 0x x 5 or 5x x i
If a polynomial P(x) has real coefficients, and if a+bi is a zero of P(x), then its complex conjugate a-bi is also a zero of P(x)
Complex Conjugate Zeros Theorem
Find all remaining zeros given the information provided.Degree: 5, 2-4i and 3 and 7i are zeros
2+4i and -7i are also zeros
Given a zero of the polynomial, determine all other zeros and write the polynomial as the product of linear factors
Conjugates
4 3 27 9 18 3 is a zeroP x x x x xi
2
3 is also zeroTherefore, 3 and 3 are factors
3 3 9
ix i x i
x i x i x
Conjugates 4 3 2 27 9 18, 9P x x x x x x
Conjugates 4 3 2 27 9 18, 9P x x x x x x
4 3 2 2 27 9 18 9 2x x x x x x x
4 3 2 2 27 9 18 9 2x x x x x x x
4 3 2 27 9 18 9 2 1x x x x x x x
ZEROS 3i 2 1
3 3 2 1P x x i x i x x
HOMEWORK: Read 382-387, p388 1-21 odd
HOMEWORK
Pages 388; 1-21 odd