The Fundamental Theorem of Algebra

Index card: 33a

Suppose a polynomial equation is pulled out of the air—perhaps this one:

Is it guaranteed to have a solution? In other words, must there exist a value of $\,x\,$ for which it is true?

Alternatively, re-arrange the equation above by getting a zero on the right-hand side. Take the resulting expression on the left of the equation, and use it to define a function $\,f\,$:

Is this function $\,f\,$ guaranteed to have a zero? In other words, must there exist a value of $\,x\,$ for which $\,f(x)\,$ is zero?

These are precisely the questions for which the Fundamental Theorem of Algebra provides a beautiful answer!

Does a Polynomial Equation Always Have a Solution?

Short Answer: It depends! What kind of solutions are you looking for?

If you're looking specifically for real number solutions, then the answer to the question ‘Does a polynomial equation always have a solution?’ is no.

For example, the polynomial equation $\,x^2 = -1\,$ doesn't have any real number solutions. Why not? Because every real number, when squared, is greater than or equal to zero (hence can't possibly equal $\,-1\,$).

If you're looking for complex number solutions (which include any real number solutions), then you're in luck. Indeed, the Fundamental Theorem of Algebra tells us that there always exists a solution, as long as you look in the set of complex numbers.

(By the way, both $\,i\,$ and $\,-i\,$ (where $\,i = \sqrt{-1}\,$) are complex number solutions of the equation $\,x^2 = -1\,.$ )

The Fundamental Theorem of Algebra

The statement of the Fundamental Theorem of Algebra is short and simple. Don't let its simplicity fool you—it is a very powerful result.

Recall that the symbol $\Bbb C\,$ (blackboard bold C) represents the set of complex numbers, and the symbol $\Bbb R\,$ (blackboard bold R) represents the set of real numbers.

Comments on the Fundamental Theorem of Algebra

• A ‘constant polynomial’ is so-named because its output never changes—it is constant.

  For example, the function $\,f\,$ defined by $\,f(x) = 5\,$ is a constant polynomial. No matter what the input is, the output is always $\,5\,.$

• Most constant polynomials have no zeros.

  For example, consider $\,f(x) = 5\,.$ Is $\,f(x)\,$ ever equal to zero? No—it's always $\,5\,.$

• On the other hand, the constant function $\,g\,$ defined by $\,g(x) = 0\,$ (the zero function) is always zero. It has infinitely many zeros.
• For these reasons, the Fundamental Theorem of Algebra only deals with non-constant polynomials.

• Remember that the set of complex numbers includes the real numbers. In other words, if $\,x\in\Bbb R\,,$ then $\,x\in\Bbb C\,.$

  That is, every real number is also a complex number. Indeed, the real numbers are the $x$-axis (the real axis) in the complex plane.

• There are loads of complex numbers that are not real numbers. Any number that doesn't lie on the real axis in the complex plane isn't a real number.
• A ‘polynomial with complex coefficients’ allows terms of the form $\,ax^n\,,$ where $\,a\in\Bbb C\,.$

• Is a polynomial like $\,P(x) = 5x^3 - 2x + 1\,$ a ‘polynomial with complex coefficients’?

  Yes! Every real number is a complex number. The Fundamental Theorem of Algebra covers (in particular) all polynomials with real coefficients.

• Recall that ‘at least one’ means greater than or equal to one.

• The Fundamental Theorem of Algebra is an existence theorem.

  It guarantees that there exists a complex number $\,c\,$ for which $\,P(c) = 0\,,$ as long as the function $\,P\,$ is a non-constant polynomial with complex coefficients.

• Recall that a corollary is an interesting consequence of a theorem—usually something that is easy to prove.
• The positive integers are: $\,1, 2, 3, \ldots$

• Consider the polynomial $\,P\,$ defined by: $$P(x) = (x-1)^2(x-5)^4(x+9)$$

  It has degree $\,7\,.$ If it were multiplied out, the highest power on $\,x\,$ would be $\,7\,.$

  How many zeros does it have? This question is a bit ambiguous, so needs some clarification, as follows:

• The polynomial $\,P\,$ has three distinct (that is, different) zeros: $\,1\,,$ $\,5\,,$ and $\,-9\,.$

  If you look at the graph of $\,P\,,$ it crosses the $x$-axis at exactly these three places.

• The polynomial $\,P\,$ has seven zeros, counting multiplicities:

  • the zero $\,1\,$ has multiplicity $\,2\,$ (from the two factors of $\,(x-1)\,,$ written as $\,(x-1)^2\,\ $)
  • the zero $\,5\,$ has multiplicity $\,4\,$ (from the four factors of $\,x-5\,,$ written as $\,(x-5)^4\,\ $)
  • the zero $\,-9\,$ is a simple zero (multiplicity $\,1\,$) (from the factor $\,(x+9)\,\ $)

  Counting multiplicities, the seven zeros of $\,P\,$ are: $$1\,,\ 1\,,\ 5\,,\ 5\,,\ 5\,,\ 5\,,\ -9$$

• ‘Counting multiplicities’ means that if $\,c\,$ is a zero of multiplicity $\,k\,,$ then $\,c\,$ is counted as a zero $\,k\,$ times.

How is This Corollary an Easy Consequence of the Fundamental Theorem of Algebra?

Here's the idea:

• Start with a polynomial with complex coefficients that has degree $\,n\,.$ For convenience, call this polynomial $\,P\,$ and let $\,z\,$ represent a typical input.
• By the Fundamental Theorem of Algebra, $\,P\,$ has a zero in $\,\Bbb C\,.$ Call this zero $\,z_1\,.$

• Since $\,z_1\,$ is a zero of $\,P\,,$  $\,z - z_1\,$ is a factor of $\,P(z)\,.$

  Thus, $$P(z) = (z-z_1)P_1(z)\,,$$ where $\,P_1\,$ is a polynomial with complex coefficients that has degree $\,n-1\,.$

• By the Fundamental Theorem of Algebra, $\,P_1\,$ has a zero in $\,\Bbb C\,.$ Call this zero $\,z_2\,.$

• Since $\,z_2\,$ is a zero of $\,P_1\,,$  $\,z - z_2\,$ is a factor of $\,P_1(z)\,.$

  Thus, $$P_1(z) = (z-z_2)P_2(z)\,,$$ where $\,P_2\,$ is a polynomial with complex coefficients that has degree $\,n-2\,.$

  Combining results thus far: $$P(z) = (z - z_1)(z - z_2)P_2(z)$$

• Repeat as needed.

  Eventually, we have $$P(z) = (z - z_1)(z - z_2)\cdots(z - z_n)P_n(z)$$

  where $\,P_n(z)\,$ is a polynomial with complex coefficients that has degree $\,n - n = 0\,.$

• Since $\,P_n(z)\,$ has degree zero, it is a constant. Call it $\,K\,$ for simplicity.

• Thus: $$ P(z) = K(z - z_1)(z - z_2)\cdots(z - z_n)$$ for some complex constant $\,K\,.$

• Thus, $\,P\,$ has the $\,n\,$ zeros $\,z_1\,,\,z_2\,,\,\ldots \,,\,z_n\,.$ Some of the zeros $\,z_i\,$ may be identical, which is why we need to count multiplicities.
• Voila!