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5.6 Complex Zeros; Fundamental Theorem of Algebra

In this section, we will study the following topics:

Conjugate Pairs Theorem

Finding a polynomial function given specified zeros

Finding complex zeros of a polynomial

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Review: Complex Numbers(pp 109-114)

The set of COMPLEX NUMBERS includes all real and imaginary numbers.

The imaginary unit, i, is defined as

Thus, we have

Complex numbers are of the form:

where and are real numbers.a bi a b

1.i

2 1i

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Complex Zeros of a Polynomial Function

In section 5.1, we saw that an nth-degree polynomial could have AT MOST n real zeros. In this section, we will do better than that.

The following result is derived from the all-important FUNDAMENTAL THEOREM OF ALGEBRA.

In the complex number system, a polynomial of degree n (n ≥ 1) has exactly n complex zeros (though not necessarily distinct*).

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Complex Zeros of a Polynomial Function

The zeros may be all real, all imaginary, or a combination.

It depends upon the degree of the polynomial and the individual function.

For example, the cubic polynomial function f(x) = (x – 2)3 has a triple zero at x = 2.

*Note that the number of zeros includes repeated zeros. In other words, a double zero counts as two zeros, a triple zero counts as three zeros, etc.

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Complex Zeros of a Polynomial Function

Now that we can determine the number of zeros, we need to actually find those zeros.

We will use the same techniques as we did in sect. 5.5 to find the real zeros for higher-degree polynomials:

Use the Rational Zero Theorem and the graph to locate one or more rational zeros

Use synthetic division to find the depressed equation (keep dividing until the depressed equation is quadratic)

Use factoring, extracting the roots, or quadratic formula to solve the resulting quadratic equation

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Complex Zeros of a Polynomial Function

Example#1* 3 2Find ALL of the zeros of ( ) 3 2 27 18 and write

in factored form.

f x x x x f

Solution:

By the F.T. of A., we know that f has ____ complex zeros.

Start by listing all of the potential rational zeros:

potential rational zeros=

p

q

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Complex Zeros of a Polynomial Function

Example #1 (continued)

We can now either sub each of the possible (+) rational zeros into the polynomial or use synthetic division, until we find an actual rational zero.

OR

We can use our graphing calculators to help us locate one rational zero and then verify it is an actual zero using synthetic division.

From the graph it looks like the rational zero is about ________.

Use synthetic division to verify that _______ is actually a zero.

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Complex Zeros of a Polynomial Function

Example #1* (cont)

3 2 27 18The remainder is ____, therefore x = ________

is a zero.

The depressed equation is _____________ = 0

Solve this quadratic equation for x to find the remaining two zeros.

So, the zeros of f(x) are ___________________________.

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Complex Zeros of a Polynomial Function

Example #1 (continued)

The factored form of f(x) is

( )

Cleaning this up a bit, we have:

( )

f x x x x

f x

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Complex Zeros of a Polynomial Function

Example #2

4 3 2Find ALL of the zeros of ( ) 2 +5x 4 5 2 and write

in factored form.

f x x x x f

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Complex Zeros of a Polynomial Function

Example #2 (cont)

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Complex Zeros of a Polynomial Function

Example #3

3 2Find ALL of the zeros of ( ) x 3 15 125.f x x x

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Complex Zeros of a Polynomial Function

Example #3 (cont)

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Conjugate Pairs Theorem

We can work backwards to find a polynomial with specified zeros.

But first……

Conjugate Pairs Theorem

Let f(x) be a polynomial whose coefficients are REAL numbers.

If is a zero of the function, the CONJUGATE is ALSO a zero of the function.

e.g. If you know that – 3 + i is a zero of a given polynomial function (with real coefficients), you also know that _________ is a zero.

Complex zeros occur in conjugate pairs!!

r a bi r a bi

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Finding a Polynomial Function with Given Zeros

Example #1*

Find a cubic polynomial with real coefficients that has zeros -1 and 6 + 5i

Write each zero in factored form.

Distribute the negative to remove inner parentheses.

Multiply the trinomials.

Multiply the binomial x trinomial.

Solution:

We know that _______________ is also a zero.

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Finding a Polynomial Function with Given Zeros

Example #2

Find a quartic polynomial with integer coefficients that has zeros 5, 5, 3 i

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Using Complex Conjugates to Find All Zeros

Here’s a neat example of how you can use the fact that complex zeros are conjugates to find all of the zeros of a polynomial function.

Example*

3 2Find all of the zeros of ( ) 4 14 20 given that

1 3 is a zero of .

f x x x x

i f

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Using Complex Conjugates to Find All Zeros

Solution:

1. Use the fact that _________ is a zero and write each complex zero in factored form. Multiply the factors.

So, the zeros of f(x) are ___________________________

2. Divide this product into f to obtain the remaining factor.

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End of Section 5.6