Multiply the factors and write in standard form.1)
2)
3)
Solve each equation in the complex number system:4)
5)
6)
))(( ixix
)2)(2( ixix
))3())(3(( ixix
Review – Complex Numbers Name: _________________
10 2 x
xx 40 3
36130 24 xx
Multiply the factors and write in standard form.1)
2)
3)
Solve each equation in the complex number system:4)
5)
6)
))(( ixix
)2)(2( ixix
))3())(3(( ixix
Review – Complex Numbers Name: _________________
10 2 x
xx 40 3
36130 24 xx
1. How many zeros does this function have?
1)( 2 xxf
I. Complex Zeros
2. How many zeros does this function have?
xxxf 4)( 3
Complex Zeros
3. How many zeros does this function have?
3613)( 24 xxxf
An example with no real zeros
How can you be sure there are no real zeros when you graph it on the calculator?
Section 3.7 Complex Zeros
1.Complex Zeros
2.Factoring with Complex Zeros
3.Conjugate Pairs Theorem
4.Finding Complex Zeros
• A polynomial of degree n will have exactly• n zeros
• and • factors in the complex plane
I. Complex ZerosFundamental Theorem of Algebra
)())(()( 21 nrxrxrxaxf
II. Factoring with real zerosFind the zeros and write as a product of linear factors.
20243132)( 234 xxxxxf
Example: All real and rational zeros
Find rational zeros on calculator first.
II. Factoring with complex zerosFind the zeros and write as a product of linear factors.
f (x) x 4 13x 2 36
Example: No real zeros (all pure imaginary zeros)
Example: real and complex zeros. Factor completely4 3 2( ) 3 19 27 252f x x x x x
Calculator shows rational zeros are at: 4 and -7
Find rational zeros first.
Example: real, irrational, complex zeros
Calculator shows rational zeros are at: 4 and -3
𝑥6−𝑥5 −6 𝑥4 − 4 𝑥2− 4 𝑥+24
III. Conjugate Pairs Theorem
1)( 1. 2 xxf
xxxf 4)( 2. 3
Complex zeros always come in conjugate pairs!
If a + bi is a zero of f, then a – bi is also a zero of f
Determine the zeros
III. a) Corollary to Conjugate Pairs TheoremSuppose a function has degree and zeros as given, what are the remaining zeros?
1. Degree 3; zeros: 1, 2 + i2. Degree 6; zeros 3 + 2i, i, -4 + i
Can a polynomial of degree 3 have as zeros 2i, 4-i ?
A polynomial f of odd degree with real coefficients has at least one real zero.
Will a polynomial of degree 4 have real zeros if it has complex zeros 4-i, and 5i ?
IV. Linear Factors of Complex Zeros
Form a polynomial with real coefficients satisfying :
1) degree 4 and zeros at: 3, multiplicity 2; and
2) degree 3 and zeros at: 4, and
i2
22i
V. Given a complex zero, find remaining Complex Zeros
Suppose the function has as a zero.Determine the remaining zeros of the function.
106116)( 234 xxxxxf
i3
a) Determine the matching pair of complex zeros.b) Multiply linear factors of the complex zeros.c) Polynomial division.
d) Solve for zeros of q(x)
)((b)part in result
)( xqxf
VI. Find Complex Zeros and write in factored form
Suppose the function has as a zero. Write in factored form:
18452553)( 234 xxxxxf
i3
)())(()( 21 nrxrxrxxf
VI. Find Complex Zeros and write in factored form
(p. 237 #29). Suppose the function has as a zero.Determine the remaining zeros of the functionWrite in factored form
352528101523)( 2345 xxxxxxf
i4
