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# Fundamental Theorem of Algebra

## Find all zeroes of a polynomial, including complex solutions

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Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is really the foundation on which most study of Algebra is built. In simple terms it says that every polynomial has 'zeroes'. That means that every polynomial can be factored and set equal to zero (the Factorization Theorem).

That is an extremely broad statement! Every polynomial can be factored? What about functions like ? What about crazy big ones, like ?

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### Guidance

Here are four important theorems in the study of complex zeros of polynomial functions:

The Fundamental Theorem of Algebra

If is a polynomial of degree , then has at least one zero in the complex number domain. In other words, there is at least one complex number such that .

There is no rigorous proof for the fundamental theorem of algebra. Some mathematicians even believe that such proof may not exist. However, the theorem is considered to be one of the most important theorems in mathematics. A corollary of this important theorem is the factorization theorem,

Theorem 2: The Factorization Theorem

If

where , and is a positive integer, then

where the numbers are complex numbers.

Theorem 3: The Roots Theorem

If is a polynomial of degree , where , then has, at most, zeros.

Notice that this theorem does not restrict that the zeros must be distinct. In other words, multiplicity of the zeros is allowed. For example, the quadratic equation has one zero, -3, and we say that the function has -3 as a double zero or one zero with multiplicity . In general, if

then is a zero of the polynomial and of multiplicity . For example,

has 2 as one zero with and -5 as a zero with .

Theorem 4: Conjugate Pairs Theorem

If is a polynomial of degree , with and with real coefficients, and if , where , then . Where is the complex conjugate of .

This is a fascinating theorem! It says basically that if a complex number is a zero of a polynomial, then its complex conjugate must also be a zero of the same polynomial. In other words, complex roots (or zeros) exist in conjugate pairs for the same polynomial. For example, the polynomial function

has two zeros: one is the complex number . By the conjugate root theorem, is also a zero of . We can easily prove that by multiplication:

#### Example A

Write as a complex polynomial in factored form.

Solution

Notice that has no real roots. You can see this in the graph of , or by looking at the discriminant, .

Using the quadratic formula, the roots of are

Finally, writing in factored form,

#### Example B

What is the form of the polynomial if it has the following numbers as zeros: and ?

Solution

Since the numbers and are zeros, then they are roots of . It follows that they must satisfy the conjugate root theorem. Thus and must also be roots to . Therefore,

Simplifying,

After multiplying we get,

which is a fifth degree polynomial. Notice that the total number of zeros is also 5.

#### Example C

What is the multiplicity of the zeros to the polynomial

Solution

With the help of the rational zero theorem and the synthetic division, we find that is a zero of ,

Using synthetic division on the quotient, we find that 3 is again a zero:

or from the Root Theorem (Theorem 3), we write our solution as

So 3 is a double zero and and are each of .

### Vocabulary

Multiplicity describes the number of times a given term may apply as a zero of a given function.

A Fundmental theorem is a theory upon which many other theories and rules are built.

Synthetic division is a shorthand version of polynomial long division.

### Guided Practice

Problems 1 - 3: Identify or estimate the values of the Zeroes from the graphs or equations and state their multiplicities

1)

2) A 4th degree equation:

3)

Problems 4 - 5: Find a polynomial function with real coefficients that has the given numbers as its zeros.

4)

5)

Solutions

1) To identify the roots and their multiplicities:

First set the function equal to 0:
The roots then are and
Since the root appears twice, it has a multiplicity of 2, whereas the root appears only once: multiplicity 1.

Note: The graph of this function (shown below) will pass through the axis at the root and bounce off the axis at the root

If a root has an even multiplicity, it will "bounce" off of the axis, and if it has an odd multiplicity, it will pass through.

2) Recall that the roots are locations where the graph contacts the x axis. The image indicates this happens at x = -3, -2, and 1.

Applying the rule from the solution to question 1 tells us that the root "-3" has an even multiplicity, since it bounces off of the axis. The other 2 roots have odd multiplicities, since they pass through.
The question specifies that this is a 4th degree equation, therefore the root "-3" has a multiplicity of 2 and the other two roots each display multiplicity 1.

3) First, factor the polynomial:

The roots are and
The multiplicities stem from the multiples of the same binomial, so the root is multiplicity 3 and is multiplicity 2.
A graph of this equation would show the line passing through and bouncing off of

4) To find a function with the specified Zeroes:

Recall that the zeroes of a function are the additive inverse of the constant term in each binomial of the factored polynomial, giving:
Distribute
Multiply the polynomials

is the specified polynomial.

5) To identify a function with the given Zeroes:

Write out binomials with additive inverses of the given Zeroes:
Recall that all complex Zeroes come in pairs, meaning that has a conjugate: Giving:
Distribute:
Distribute again (I chose to distribute the latter two binomials first):
Multiply the polynomials:

is the polynomial

### Practice

In problems 1-5, find a polynomial function with real coefficients that has the given numbers as its zeros.

1. If is a root of the polynomial , find all other roots of .
2. If is a zero of the polynomial , find all other zeros of .

In Problems 8-10, determine whether the given number is a zero of the given polynomial. If so, determine its multiplicity.

For problems 11 - 15, sketch the graph, properly indicating multiplicities.

### Vocabulary Language: English

Complex Conjugate

Complex Conjugate

Complex conjugates are pairs of complex binomials. The complex conjugate of $a+bi$ is $a-bi$. When complex conjugates are multiplied, the result is a single real number.
complex number

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.
conjugate pairs theorem

conjugate pairs theorem

The conjugate pairs theorem states that if $f(z)$ is a polynomial of degree $n$, with $n\ne0$ and with real coefficients, and if $f(z_{0})=0$, where $z_{0}=a+bi$, then $f(z_{0}^{*})=0$. Where $z_{0}^{*}$ is the complex conjugate of $z_{0}$.
fundamental theorem of algebra

fundamental theorem of algebra

The fundamental theorem of algebra states that if $f(x)$ is a polynomial of degree $n\ge 1$, then $f(x)$ has at least one zero in the complex number domain. In other words, there is at least one complex number $c$ such that $f(c)=0$. The theorem can also be stated as follows: an $n^{th}$ degree polynomial with real or complex coefficients has, with multiplicity, exactly $n$ complex roots.
Imaginary Number

Imaginary Number

An imaginary number is a number that can be written as the product of a real number and $i$.
Imaginary Numbers

Imaginary Numbers

An imaginary number is a number that can be written as the product of a real number and $i$.
Multiplicity

Multiplicity

The multiplicity of a term describes the number of times the given term acts as a zero of the given function.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Roots

Roots

The roots of a function are the values of x that make y equal to zero.
Synthetic Division

Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.
Zero

Zero

The zeros of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.
Zeroes

Zeroes

The zeroes of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.