“Is it better to be feared or respected? And I
say, is it too much to ask for both?”
3.7: Complex
Zeros
Fundamental Theorem of Algebra
If f (x) is a polynomial of degree n > 0, then f has exactly
n linear factors and therefore n zeros in the complex
plane a+bi.
f (x)  x 4  3x 3 19x 2  27x  252
f ( x)  x3  13x 2  36

f ( x)  x6  x5  2 x 4  3x3  40 x 2  137 x  58
Complex Zeros
Find the complex zeros of the polynomial function, and write
as a product of linear factors.
f (x)  x 4  13x 2  36
Complex Zeros
Find the complex zeros of the polynomial function, and write
as a product of linear factors.
f (x)  x 4  3x 3 19x 2  27x  252
Complex Zeros
Find the complex zeros of the polynomial function, and write
as a product of linear factors.
f ( x)  3x 4  2 x3  33x 2  82 x  40
Conjugate Pairs Theorem
Let f (x) be a complex polynomial whose coefficients are
real numbers. If r = a + bi is a zero of f, then the
complex conjugate r  a  bi is also a zero of f.
Find a polynomial f of degree 4 in standard form whose
coefficients are real numbers and has zeros 1, 2, and 2 + i.

Complex Zeros
Use the given zero to find the remaining zeros of the given
function.
f (x)  2x 4  x 3  7x 2  4 x  4; zero : 2i
Complex Zeros
Use the given zero to find the remaining zeros of the given
function.
f (x)  x 4  9x 2  14 x  30; zero : 1 3i
“Don’t worry Wilson, I’ll do all the
paddling. You just hang on!”
3.7 Complex Zeros
Homework #10:
Page 237
#31-39 Odd
3.7 Complex Zeros
Homework #11:
Page 237
#7-29 Odd