COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

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COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

Transcript of COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

Page 1: 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?

Page 2: COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why 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

Page 3: COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

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

Page 4: COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

Theorem

The fundamental theorem of algebra was proved by Karl Friedrich Gauss (1777 – 1855) when he was 22 years old.

Page 5: COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

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

Page 6: COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

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)

Page 7: COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

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


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