4.5 The Fundamental Theorem of Algebra (1 of 2) Apply the fundamental theorem of algebra, number of zeros theorem, and conjugate zeros theorem. Factor polynomials having complex zeros Solve polynomial equations having complex solutions

Fundamental Theorem of Algebra (2 of 2) A polynomial f(x) of degree n, with n ≥ 1, has at least one complex zero. The fundamental theorem of algebra guarantees that every polynomial has a complete factorization, if we are allowed to use complex numbers.

Number of Zeros Theorem A polynomial of degree n, with n ≥ 1 has at most n distinct zeros.

Example: Classifying zeros (1 of 4) All zeros for the given polynomials are distinct. Use the figures to determine graphically the number of real zeros and the number of nonreal complex (imaginary) zeros.

Example: Classifying zeros (2 of 4) The graph of ƒ(x) crosses the x-axis once, so there is one real zero. Since ƒ is degree 3 and all zeros are distinct, there are two nonreal complex zeros. Solution Real zeros correspond to x-intercepts; complex nonreal (imaginary) zeros do not.

Example: Classifying zeros (3 of 4) The graph of g(x) never crosses the x-axis. Since g is degree 2, there are no real zeros and two nonreal complex zeros.

Example: Classifying zeros (4 of 4) The graph of h(x) is shown. Since h is degree 4, there are two real zeros and the remaining two zeros are nonreal complex.

Example: Constructing a polynomial with prescribed zeros Determine a polynomial f(x) of degree 4 with leading coefficient 2 and zeros −3, 5, i and −i in (a) complete factored form and (b) expanded form. Solution a. Let an = 2, c1 = −3, c2 = 5, c3 = i, and c4 = −i. f(x) = 2(x + 3)(x − 5)(x − i)(x + i)

Conjugate Zeros Theorem If a polynomial f(x) has only real coefficients and if a + bi is a zero of f(x), then the conjugate a − bi is also a zero of f(x).

Example: Finding complex zeros of a polynomial (1 of 3) Solution By the conjugate zeros theorem it follows that i must be a zero of f(x). (x + i) and (x − i) are factors (x + i)(x − i) = x2 + 1, using long division we can find another quadratic factor of f(x).

Example: Finding complex zeros of a polynomial (2 of 3) Long division

Example: Finding complex zeros of a polynomial (3 of 3) Use the quadratic formula to find the zeros of x² + x + 1.

Polynomial Equations with Complex Solutions

Example: Solving a polynomial equation (1 of 3) Solve x³ = 3x² − 7x + 21. Solution Rewrite the equation as f(x) = 0, where f(x) = x³ − 3x² + 7x − 21. We can use factoring by grouping or graphing to find one real zero of f(x). Let’s use graphing in this example.

Example: Solving a polynomial equation (2 of 3) The graph shows a zero at 3. So, x − 3 is a factor. Now use synthetic division by 3.

Example: Solving a polynomial equation (3 of 3)