SECTION 3.6SECTION 3.6

COMPLEX ZEROS; COMPLEX ZEROS; FUNDAMENTAL THEOREM OF FUNDAMENTAL THEOREM OF

ALGEBRAALGEBRA

COMPLEX POLYNOMIAL FUNCTION

COMPLEX POLYNOMIAL FUNCTION

A complex polynomial function A complex polynomial function ff of degree n is a complex function of degree n is a complex function of the formof the form

f(x) = a f(x) = a n n x x nn + a + a n-1n-1 x x n-1n-1 + . . . + a + . . . + a11x x + a+ a00

where awhere ann, a , a n-1n-1, . . ., a, . . ., a11, a, a00 are are complex numbers, acomplex numbers, ann 0, n is a 0, n is a nonnegative integer, and x is a nonnegative integer, and x is a complex variable.complex variable.

COMPLEX ZEROCOMPLEX ZERO

A complex number A complex number rr is called a is called a complex zero of a complex complex zero of a complex function function ff if f(r) = 0. if f(r) = 0.

COMPLEX ZEROSCOMPLEX ZEROS

We have learned that some We have learned that some quadratic equations have no real quadratic equations have no real solutions but that in the complex solutions but that in the complex number system every quadratic number system every quadratic equation has a solution, either equation has a solution, either real or complex.real or complex.

FUNDAMENTAL THEOREM OF ALGEBRA

FUNDAMENTAL THEOREM OF ALGEBRA

Every complex polynomial Every complex polynomial function f(x) of degree n function f(x) of degree n 1 has 1 has at least one complex zero.at least one complex zero.

THEOREMTHEOREM

Every complex polynomial Every complex polynomial function f(x) of degree n function f(x) of degree n 1 can 1 can be factored into n linear factors be factored into n linear factors (not necessarily distinct) of the (not necessarily distinct) of the formform

f(x) = af(x) = ann(x - r(x - r11)(x - r)(x - r22)) (x - r (x - rnn))

where awhere ann, r, r11, r, r22, . . ., r, . . ., rnn are are complex numbers.complex numbers.

CONJUGATE PAIRS THEOREM

CONJUGATE PAIRS THEOREM

Let f(x) be a complex Let f(x) be a complex polynomial whose coefficients polynomial whose coefficients are real numbers. If r = a + bare real numbers. If r = a + bii is a zero of f, then the is a zero of f, then the complex conjugate r = a - bcomplex conjugate r = a - bii is is also a zero of f.also a zero of f.

CONJUGATE PAIRS THEOREM

CONJUGATE PAIRS THEOREM

In other words, for complex In other words, for complex polynomials whose coefficients polynomials whose coefficients are real numbers, the zeros are real numbers, the zeros occur in conjugate pairs.occur in conjugate pairs.

CORORLLARYCORORLLARY

A complex polynomial f of odd A complex polynomial f of odd degree with real coefficients degree with real coefficients has at least one real zero.has at least one real zero.

EXAMPLEEXAMPLE

A polynomial f of degree 5 A polynomial f of degree 5 whose coefficients are real whose coefficients are real numbers has the zeros 1, 5i, numbers has the zeros 1, 5i, and 1 + i. Find the remaining and 1 + i. Find the remaining two zeros.two zeros.

- 5i- 5i

1 - i1 - i

EXAMPLEEXAMPLE

Find a polynomial f of degree 4 Find a polynomial f of degree 4 whose coefficients are real whose coefficients are real numbers and has the zeros 1, numbers and has the zeros 1, 1, and - 4 + i.1, and - 4 + i.

f(x) = a(x - 1)(x - 1)[x - (- 4 + i)][x f(x) = a(x - 1)(x - 1)[x - (- 4 + i)][x - (- 4 - i)]- (- 4 - i)]

First, let a = 1; Graph the First, let a = 1; Graph the resulting polynomial. Then look resulting polynomial. Then look at other a’s.at other a’s.

EXAMPLEEXAMPLE

It is known that 2 + i is a zero It is known that 2 + i is a zero of of

f(x) = xf(x) = x44 - 8x - 8x33 + 64x - 105 + 64x - 105

Find the remaining zeros.Find the remaining zeros.

- 3, 7, 2 + i and 2 - i- 3, 7, 2 + i and 2 - i

EXAMPLEEXAMPLE

Find the complex zeros of the Find the complex zeros of the polynomial functionpolynomial function

f(x) = 3xf(x) = 3x44 + 5x + 5x33 + 25x + 25x22 + 45x - + 45x - 1818

CONCLUSION OF SECTION 3.6CONCLUSION OF SECTION 3.6