COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?
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Transcript of COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?
COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRAWhy do we have to know imaginary numbers?
Fundamental Theorem of Algebra Every complex polynomial function f(x)
of degree n ≥ 1 has at least one complex zero.
Explanation from the expert
Theorem
Every complex polynomial function f(x) of degree
n ≥ 1 can be factored into n linear factors (not necessarily distinct) of the form
1 2
1 2
( ) ( )( ) . . . ( )
where , , , . . are complex numbers. That is every complex
polynomial function of degree 1 has exactly (not necessarily
distinct) zeros.
n n
n n
f x a x r x r x r
a r r
n n
r
Theorem
The fundamental theorem of algebra was proved by Karl Friedrich Gauss (1777 – 1855) when he was 22 years old.
Conjugate Pairs Theorem
Let ( ) be a polynomial whose coefficients are real numbers. If
is a zero of , then the complex conjugate i also
a zero of .
f x
r a bi f r a bi
f
Finding a Polynomial Whose Zeros are Given Extra Examples Find a polynomial f of degree 4 whose
coefficients are real numbers and that has the zeros 1, 1, and
-4 + i.(x – 1) (x – 1) (x – (-4 + i)) (x – (-4 – i))Do FOIL on the conjugate pairs first. (x^2 –(-4 – i)x – (-4 + i)x +(-4 + i)(-4 – i))=x^2 +4x – ix +4x +ix +16 + 4i – 4i – i^2= x^2 + 8x + 17 (Remember i^2 = -1)
Finding a Polynomial Whose Zeros are Given Now multiply the other two zeros together
using FOIL:
(x – 1) (x – 1) = x^2 – 2x + 1 Finally multiply the two polynomials together
using the distributive property
(x^2 – 2x + 1) (x^2 + 8x + 17)
X^4 + 8x^3 + 17x^2
-2x^3 – 16x^2 – 34x
x^2 + 8x + 17=
X^4 + 6x^3 + 2x^2 – 26x + 17