Section 2.5 – Complex Zeros The Fundamental Theorem of Algebra

Section 2.5 – Complex ZerosThe Fundamental Theorem of Algebra We put the “FUN” in FUN The damental Theorem of Algebra

Complex Zeros Fundamental Theorem of Algebra: If f(x) is a p.f. of degree n, then f(x) has n complex roots (real and nonreal). Some of these zeros may be repeated. Linear Factorization Theorem: If f(x) is a p.f. of degree n>0, then f(x) has precisely n linear factors and where a is the leading coefficient of f(x) and z1, z2, … zn are the complex zeros of f(x). The zeros, zi, are not necessarily distinct numbers. Some may be repeated.

Complex Conjugate Zeros • Suppose that f(x) is a p.f. with real coefficients. If a and b are real numbers with b 0, and a + bi is a zero of f(x), then its complex conjugate, a – bi, is also a zero of f(x). • Simply stated, complex roots occur in conjugate pairs.

SUMMARY • Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients. • Every polynomial function of odd degree with real coefficients has at least one real zero.

Checklist: • How many total zeros - include imaginary and repeated zeros counted individually? • How many positive REAL zeros (pos x-intercepts) by DesCcartes? • How many negative REAL zeros (negx-intercepts) by DesCartes)? • Possible RATIONAL Zeros? 5 5 = degree of f(x) Sum of coefficients is 0, so 1 is a zero Now try 2. There are 5 sign changes in f(x)… so there can be either 5, 3 or 1 positive REAL zeros. There are 0 sign changes in f(x)… so there are NO negative REAL zeros. Now try 3. All non-negative, so 3 is the upper bound of real zeros.

Use Q.F. to find remaining two zeros Summary: 3 positive Real (All rational) 2 imaginary zeros

Section 2.5 – Complex Zeros The Fundamental Theorem of Algebra