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Fundamental Theorem of Algebra

If you're just encountering the fundamental theorem of algebra for the first time, you might be wondering why this is the first time you've heard of it. After all, you might have been studying algebra for quite some time. The "fundamental theorem of algebra" sounds pretty important.. So why are you only hearing about it now? That being said, this theorem does provide us with interesting information about virtually all polynomials. Let's find out more:

What does the fundamental theorem of algebra say?

The fundamental theorem of algebra states that:

  • Any non-constant, single-variable polynomial of degree n with complex coefficients has exactly n complex roots, counting multiplicities.

But there is also another definition of the theorem of algebra:

  • A polynomial function with complex numbers for coefficients has at least one zero in the set of complex numbers.

We also need to note that real numbers are a subset of the complex numbers. This is because every real number can be written in the format a+bi, with b equaling zero. In other words, this theorem is also true for polynomials with real coefficients.

We might also state the fundamental theorem of algebra in a different way:

  • Every non-constant, single-variable polynomial with complex coefficients has at least one complex root.

But this is just another way of stating the earlier, second definition.

Why is the fundamental theorem of algebra useful?

But why exactly do we need to know the fundamental theorem of algebra?

One of the most useful things about the fundamental theorem of algebra is that it can help us determine how many potential roots a polynomial might have.

You may recall that the "root" or "zero" of a polynomial is the value of the variable that makes the polynomial zero. In other words, the root is what we need to plug into our equation so that the output is zero.

Here's why the fundamental theorem of algebra is useful in this situation:

f x = 4 x 3 + 2 x 2 - 7

We know that the highest value of a variable's exponent is its degree. Here we can see that the polynomial is degree 3. This means that there are three roots or zeros. If we were to visualize these roots on a graph, we would see where the polynomial intersects with the x-intercept. In this case, there are three locations where this occurs. These points represent the roots of the polynomial.

Working with imaginary numbers

But how does the fundamental theorem of algebra apply to complex and imaginary numbers?

Remember that a complex number takes the following form:

[ a + b i ] where a is the real part, b is the imaginary part, and i = - 1

With these kinds of equations, we are often left with pairs of solutions that are conjugates of each other. For example:

x 2 - x + 1 = 0

We can use the quadratic equation to determine that the answer can be 0.5 - 0.866 i or 0.5 + 0.866 i . Also, note that we are left with complex numbers.

Here's another example:

g ( x ) = x 3 - 2 x 2 + 9 x - 18

Right away, we see that all of our coefficients are real numbers: 3 - 2 9

We can now equate our polynomial to zero. In other words, we add "= 0" to the end of the polynomial (or set g ( x ) = 0 ). Here's what that looks like:

0 = x 2 ( x - 2 ) + 9 ( x - 2 )

0 = ( x - 2 ) ( x 2 + 9 )

We can factor this as:

0 = ( x - 2 ) ( x + 3 i ) ( x - 3 i )

x = 2 or x = -3 i or x = 3 i

Here we can see that there are three possible roots or zeros. This makes sense, as the initial polynomial was degree 3. We also see that there are two imaginary numbers or complex numbers and a single "real number."

When we have degree 3, we can have 3 real roots or 1 real root and 2 complex roots. These are the only two possible combinations.


Topics related to the Fundamental Theorem of Algebra

Graphing Polynomial Functions

Degree (of a Polynomial)

Polynomial with Complex Roots

Flashcards covering the Fundamental Theorem of Algebra

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Pair your student with a tutor who understands the fundamental theorem of algebra

The fundamental theorem of algebra is, well.. fundamental. And it's important that your student gains a solid foothold on these fundamental concepts. Without that understanding, they could encounter serious problems as they get further into algebra. The good news is that a tutor can help them practice their skills and ask plenty of questions in a 1-on-1 environment. Tutors can also present information in new ways that may be more conducive to your student's learning style. For example, flashcards may be more effective for visual learners, while verbal cues may be more effective for verbal learners. Speak with our Educational Directors today to learn more about the tutoring process. Varsity Tutors will pair your student with an effective tutor.

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