THE FUNDAMENTAL THEOREM OF ALGEBRA

Suppose a polynomial equation is pulled out of the air—perhaps this one: [beautiful math coming... please be patient] $$7x^6 - \frac 12x^4 - 3 = 2x^9 + 5x$$ 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 $\,f\,$: [beautiful math coming... please be patient] $$f(x) = 7x^6 - \frac 12x^4 - 3 - 2x^9 - 5x$$ 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.

THEOREM the Fundamental Theorem of Algebra
Every non-constant polynomial with complex coefficients has at least one zero in $\,\Bbb C\,$.
Comments on the Fundamental Theorem of Algebra:

COROLLARY to the Fundamental Theorem of Algebra
Let $\,n\,$ be a positive integer.
Every polynomial of degree $\,n\,$ with complex coefficients has exactly $\,n\,$ zeros in $\,\Bbb C\,$,
counting multiplicities.

How is this corollary an easy consequence of the Fundamental Theorem of Algebra?

Here's the idea:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
polynomials with real number coefficients:
non-real zeros must occur in complex conjugate pairs

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13
AVAILABLE MASTERED IN PROGRESS

(MAX is 13; there are 13 different problem types.)