SECTION 3.6 • COMPLEX ZEROS; • FUNDAMENTAL THEOREM OF ALGEBRA
COMPLEX POLYNOMIAL FUNCTION A complex polynomial function f of degree n is a complex function of the form f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0 where an, a n-1, . . ., a1, a0 are complex numbers, an 0, n is a nonnegative integer, and x is a complex variable.
COMPLEX ZERO A complex number r is called a complex zero of a complex function f if f(r) = 0.
COMPLEX ZEROS We have learned that some quadratic equations have no real solutions but that in the complex number system every quadratic equation has a solution, either real or complex.
FUNDAMENTAL THEOREM OF ALGEBRA Every complex polynomial function f(x) of degree n 1 has at least one complex zero.
THEOREM Every complex polynomial function f(x) of degree n 1 can be factored into n linear factors (not necessarily distinct) of the form f(x) = an(x - r1)(x - r2) (x - rn) where an, r1, r2, . . ., rn are complex numbers.
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.
CONJUGATE PAIRS THEOREM In other words, for complex polynomials whose coefficients are real numbers, the zeros occur in conjugate pairs.
CORORLLARY A complex polynomial f of odd degree with real coefficients has at least one real zero.
EXAMPLE A polynomial f of degree 5 whose coefficients are real numbers has the zeros 1, 5i, and 1 + i. Find the remaining two zeros. - 5i 1 - i
EXAMPLE Find a polynomial f of degree 4 whose coefficients are real numbers and has the zeros 1, 1, and - 4 + i. f(x) = a(x - 1)(x - 1)[x - (- 4 + i)][x - (- 4 - i)] First, let a = 1; Graph the resulting polynomial. Then look at other a’s.
EXAMPLE It is known that 2 + i is a zero of f(x) = x4 - 8x3 + 64x - 105 Find the remaining zeros. - 3, 7, 2 + i and 2 - i
EXAMPLE Find the complex zeros of the polynomial function f(x) = 3x4 + 5x3 + 25x2 + 45x - 18