5.3 Complex Zeros; Fundamental Theorem of Algebra

Example: Form a polynomial f(x) with real coefficients having the given degree and zeros.

Degree 4; Zeros -2-3i; 5 multiplicity 2

Solution:

By the Fundamental Theorem of Algebra, since the degree of the polynomial is 4 the polynomial has 4 zeros if you count multiplicity.

There are three given zeros of -2-3i, 5, 5.

The remaining zero can be found using the Conjugate Pairs Theorem. f(x) is a polynomial with real coefficients. Since -2-3i is a complex zero of f(x) the conjugate pair of -2+3i is also a zero of f(x).

Now that all the zeros of f(x) are known the polynomial can be formed with the factors that are associated with each zero.

Since f(x) has a zero of 5, f(x) has a factor of x-5

Since f(x) has a second zero of 5, f(x) has a second factor of x-5

Since f(x) has a factor of -2-3i, f(x) has a factor of x-(-2-3i)

Since f(x) has a factor of -2+3i, f(x) has a factor of x-(-2+3i)

Form the polynomial using all of the factors. The leading coefficient will remain unknown.
Multiply the factors with complex numbers. Doing so will cancel the complex numbers from the expression Distribute the minus Multiply each term in one factor by each term in the other factor simplify combine like terms
Multiply the other pair of factors
Multiply the two trinomials by multiplying each term in the first trinomial by each term in the other trinomial and then combine like terms