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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 zeros. 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 \begin{align*}x^{2} = -4\end{align*}? What about crazy big ones, like \begin{align*}x^{6} - 23x^{5} + \frac{1}{246}x^{4}-23x^{3}\end{align*}?

Fundamental Theorem of Algebra

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

The Fundamental Theorem of Algebra

If \begin{align*}f(x)\end{align*} is a polynomial of degree \begin{align*}n\ge 1\end{align*}, then \begin{align*}f(x)\end{align*} has at least one zero in the complex number domain. In other words, there is at least one complex number \begin{align*}c\end{align*} such that \begin{align*}f(c)=0\end{align*}.

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.

The Factorization Theorem

If

\begin{align*}f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}\end{align*}

where \begin{align*}a_{n} \ne 0\end{align*}, and \begin{align*}n\end{align*} is a positive integer, then

\begin{align*}f(x)=a_{n}(x-c_{1})(x-c_{2})\cdots(x-c_{0})\end{align*}

where the numbers \begin{align*}c_{i}\end{align*} are complex numbers.

The \begin{align*}n-\end{align*}Roots Theorem

If \begin{align*}f(x)\end{align*} is a polynomial of degree \begin{align*}n\end{align*}, where \begin{align*}n\ne 0\end{align*}, then \begin{align*}f(x)\end{align*} has, at most, \begin{align*}n\end{align*} 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 \begin{align*}f(x)=x^{2}+6x+9\end{align*} has one zero, -3, and we say that the function has -3 as a double zero or one zero with multiplicity \begin{align*}k=2\end{align*}. In general, if

\begin{align*}f(x)=(x-c)^{k}q(x)\quad\text{and} \quad q(c)\ne0\end{align*}

then \begin{align*}c\end{align*} is a zero of the polynomial \begin{align*}f\end{align*} and of multiplicity \begin{align*}k\end{align*}. For example,

\begin{align*}f(x)=(x-2)^{3}(x+5)\end{align*}

has 2 as one zero with \begin{align*}k=3\end{align*} and -5 as a zero with \begin{align*}k=1\end{align*}.

Conjugate Pairs Theorem

If \begin{align*}f(z)\end{align*} is a polynomial of degree \begin{align*}n\end{align*}, with \begin{align*}n\ne0\end{align*} and with real coefficients, and if \begin{align*}f(z_{0})=0\end{align*}, where \begin{align*}z_{0}=a+bi\end{align*}, then \begin{align*}f(\overline{z_{0}})=0\end{align*}. Where \begin{align*}\overline{z_{0}}\end{align*} is the complex conjugate of \begin{align*}z_{0}\end{align*}.

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

\begin{align*}f(x)=x^{2}-2x+2\end{align*}

has two zeros: one is the complex number \begin{align*}1+i\end{align*}. By the conjugate pairs theorem (also called the conjugate root theorem), \begin{align*}1-i\end{align*} is also a zero of \begin{align*}f(x)=x^{2}-2x+2\end{align*}. We can easily prove that by multiplication:

\begin{align}\left[x-(1+i)\right]\left[x-(1-i)\right] &= (x-1-i)(x-1+i)\\ &= x^2 -x +xi - x +1 - i - xi + i +1\\ &= x^2 -2x + 2 \end{align}

Examples

Example 1

Write \begin{align*}g(x)=x^{2}+x+1\end{align*} as a complex polynomial in factored form.

Notice that \begin{align*}g(x)\end{align*} has no real roots. You can see this in the graph of \begin{align*}g(x)\end{align*}, or by looking at the discriminant, \begin{align*}b^{2}-4ac=1-4=-3\end{align*}.

Using the quadratic formula, the roots of \begin{align*}g(x)\end{align*} are

\begin{align}x &= \frac{-b \pm \sqrt{b^2-4ac}}{2z}\\ &= \frac{-1 \pm \sqrt{-3}}{2}\\ &= - \frac{1}{2} + \frac{\sqrt{3}}{2}i \ or \ - \frac{1}{2} - \frac{\sqrt{3}}{2}i \end{align}

Finally, writing \begin{align*}g(x)\end{align*} in factored form,

\begin{align*}g(x)=\left[x-\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\right]\left[x-\left(-\frac{1}{2}-\frac{\sqrt{3}}{2}i\right)\right]\end{align*}

Example 2

What is the form of the polynomial \begin{align*}f(x)\end{align*} if it has the following numbers as zeros: \begin{align*}\frac{-1}{3}, 1-i\end{align*} and \begin{align*}2i\end{align*}?

Since the numbers \begin{align*}2i\end{align*} and \begin{align*}1+i\end{align*} are zeros, then they are roots of \begin{align*}f(x)=0\end{align*}. It follows that they must satisfy the conjugate root theorem. Thus \begin{align*}-2i\end{align*} and \begin{align*}1-i\end{align*} must also be roots of \begin{align*}f(x)\end{align*}. Therefore,

\begin{align*}f(x)= \left ( x+\frac{1}{3} \right )[x-(1-i)][x-(1+i)][x-(2i)][x-(-2i)]\end{align*}

Simplifying,

\begin{align*}f(x) = \left ( x+\frac{1}{3} \right )(x-1+i)(x-1-i)(x-2i)(x+2i)\end{align*}

After multiplying we get,

\begin{align*}f(x)=\frac{1}{3}(3x^{5}-5x^{4}+13x^{3}-19x^{2}+4x+4)\end{align*}

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

Example 3

Find the multiplicity of the zeros of the following polynomial:

\begin{align*}g(x)=x^{4}-6x^{3}+18x^{2}-54x+81\end{align*}

With the help of the rational zero theorem and synthetic division, we find that \begin{align*}x=3\end{align*} is a zero of \begin{align*}g(x)\end{align*},

\begin{align*} \ 3 \ \big ) \overline{1 \ -6 \ \ 18 \ -54 \ \ \ \ 81\;}\\ \quad \ \ \underline{\downarrow \ \ \ 3 \ -9 \ \ \ \ 27 \ -81}\\ \quad \ \ 1 \ -3 \ \ \ 9 \ -27 \ \ \ \ \ 0\end{align*}

\begin{align*}g(x)=x^{4}-6x^{3}+18x^{2}-54x+81=(x-3)(x^{3}-3x^{2}+9x-27)\end{align*}

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

\begin{align*} \ 3 \ \big ) \overline{1 \ -3 \ \ 9 \ -27}\\ \quad \ \ \underline{\downarrow \ \ \ 3 \ \ \ 9 \ -27}\\ \quad \ \ 1 \ \ \ \ 0 \ \ \ 9 \ \ \ \ \ 0\end{align*}

or from the \begin{align*}n-\end{align*}Roots Theorem (Theorem 3), we write our solution as

\begin{align}g(x) &= (x-3)(x-3)(x^2+9)\\ &= (x-3)^2(x-3i)(x+3i) \end{align}

So 3 is a double zero \begin{align*}(k=2)\end{align*} and \begin{align*}3i\end{align*} and \begin{align*}-3i\end{align*} are each of \begin{align*}k=1\end{align*}.

Example 4

Identify or estimate the values of the zeros from the following equation and state their multiplicities: \begin{align*}y = (x + 2)^2(x - 1)\end{align*}

To identify the roots and their multiplicities:

First, set the function equal to 0:  \begin{align*}(x + 2)(x + 2)(x - 1) = 0\end{align*} 

The roots then are \begin{align*}x = 1\end{align*} and  \begin{align*}x = -2\end{align*} 

Since the \begin{align*}x = -2\end{align*} root appears twice, it has a multiplicity of 2, whereas the \begin{align*}x = 1\end{align*} root appears only once, so its multiplicity is 1.

Note: The graph of this function (shown below) will pass through the axis at the root \begin{align*}x = 1\end{align*} and bounce off the axis at the root \begin{align*}x = -2\end{align*}.

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.

Example 5

Identify or estimate the values of the zeros from the following graph and state their multiplicities.

A 4th degree equation:

Recall that the roots are locations where the graph contacts the \begin{align*}x-\end{align*}axis. The image indicates this happens at \begin{align*}x= -3, -2, \end{align*} and \begin{align*}1\end{align*}.

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 displayed each have a multiplicity of 1.

Example 6

Identify or estimate the values of the zeros from the following equation and state their multiplicities: \begin{align*}g(x) = (x^2 + 6x + 9)(x^3 + 6x^2 + 12x + 8)\end{align*}.

First, factor the polynomial: \begin{align*}g(x) = (x + 3)^2(x + 2)^3\end{align*} 

The roots are \begin{align*}x = -2\end{align*} and  \begin{align*}x = -3\end{align*}.

The multiplicities stem from the multiples of the same binomial, so the root \begin{align*}x = -2\end{align*} has a multiplicity of 3 and \begin{align*}x = -3\end{align*} has a multiplicity of 2.

A graph of this equation would show the line passing through \begin{align*}x = -3\end{align*} and bouncing off at  \begin{align*}x = -2\end{align*}.

Example 7 

Find a polynomial function with real coefficients that has the following values as its zeros: \begin{align*}2, 3, -3, 1\end{align*}

To find a function with the specified zeros:

Recall that the zeros of a function are the additive inverse of the constant term in each binomial of the factored polynomial, giving: \begin{align*}(x - 2)(x - 3)(x + 3)(x - 1)\end{align*} 

Distribute: \begin{align*}(x^2 - 5x + 6)(x^2 +2x - 3)\end{align*} 

Multiply the polynomials: \begin{align*}(x^4 - 3x^3 -7x^2 +27x - 18)\end{align*} 

\begin{align*}\therefore (x^4 - 3x^3 -7x^2 +27x - 18)\end{align*} is the specified polynomial.

   

 

Review

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

  1. \begin{align*}1, 2, i\end{align*}
  2. \begin{align*}2, 2, 1-i\end{align*}
  3. \begin{align*}i, i, 0, 2i\end{align*}
  4. \begin{align*}1, 1, \left(1-i\sqrt{3}\right)\end{align*}
  5. \begin{align*}0, 0, 2i\end{align*}
  6. If \begin{align*}i-1\end{align*} is a root of the polynomial \begin{align*}f(x)=x^{4}+2x^{3}-4x-4\end{align*}, find all other roots of \begin{align*}f\end{align*}.
  7. If \begin{align*}-2i\end{align*} is a zero of the polynomial \begin{align*}f(x)=x^{4}+x^{3}-2x^{2}+4x-24\end{align*}, find all other zeros of \begin{align*}f\end{align*}.

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

  1. \begin{align*}f(x)=9x^{4}-12x^{3}+13x^{2}-12x+4, x=\frac{2}{3}\end{align*}
  2. \begin{align*}f(x)=x^{4}-4x^{3}+5x^{2}-4x+4, x=2\end{align*}
  3. \begin{align*}f(x)=3x^{5}-4x^{4}+2x^{3}-\frac{3}{4}x^{2}+2x+12, x=-\frac{2}{3}\end{align*}

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

  1. \begin{align*}g(x) = x^2(x - 1)(x - 3)(x + 2)(x + 4)^2\end{align*}
  2. \begin{align*}f(x) = -x^2(x - 3)^3(x - 1)(x - 2)\end{align*}
  3. \begin{align*}f(x) = -x(x - 1)(x + 2)^3\end{align*}
  4. \begin{align*}g(x) = x^3(x + 3)^4(x - 2)\end{align*}
  5. \begin{align*}f(x) = (x + 3)^2(x - 1)(x + 1)\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.14. 

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Vocabulary

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

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

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

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

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

Imaginary Numbers

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

Multiplicity

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

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

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

Synthetic Division

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

Zero

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

Zeroes

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

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