HELP ME with the FUNDAMENTAL THEOREM of ALGEBRA!!!?

Question

i have looked ALL over the place and found many articles about the Fundamental Theorem of Algebra, yet none of them seem to make any sense to me. I am only in Algebra 1 and the discriptions of the theory are too hard for me to understand. I would really appreciate if someone could explain it on MY level or give me a link to an article that explains it in an EASY WAY!! PLEASE HELP ME!!

Answer 1

The fundamental theorem of algebra says this: A polynomial of degree n has exactly n roots in the complex numbers, counted with multiplicity.

Since you're in algebra 1, I don't know how well you understand that, or even whether you've covered all the terms yet, so let me break this down for you:

"A polynomial" -- something that looks like c_n * x^n + c_(n-1) * x^(n-1) + c_(n-2) * x^(n-2)... + c₂x² + c₁x + c₀, where each of the c_k are constants. Some concrete examples are 3x³ + 4x² - 9x - 4, 5x-2, and x²-4x+4. Note that a polynomial is not required to have multiple terms -- x is a perfectly valid polynomial. On the other hand, a polynomial is only permitted to have nonnegative integer powers of the variable -- x⁻¹ and x^(1/2) are not terms that can appear in any polynomial of x. A polynomial is also required to have only finitely many terms -- the expression [k=0, ∞]∑(x^k/k!) is not a polynomial, even though for any finite n, [k=0, n]∑(x^k/k!) is a polynomial (if you haven't learned about sigma notation yet, just file this in the back of your mind for a while, you'll understand those expressions eventually).

"of degree n" -- this means that the highest power of x in the polynomial with a nonzero coefficient. So the degree of 3x³ + 4x² - 9x - 4 is 3, and the degree of 5x-2 is 1. Some technical notes: a nonzero constant, such as 12, is considered to be a polynomial of degree 0, since it can be written as 12x⁰. The constant 0 is also considered a polynomial, but does not have a degree.

"has exactly n roots... counted with multiplicity" -- A root of a polynomial is a value of x that would make the expression equal to zero. Most likely, you have learned (or will learn shortly) that a number r is a root of a polynomial if and only if (x-r) is a factor of the polynomial. The multiplicity of a root is then simply the number of times that (x-r) appears as a factor. So what this says is that if you take all the roots, counting each root r the number of times (x-r) appears as a factor, then you will have exactly n of them, where n is the degree of the polynomial.

"in the complex numbers" -- this is the important caveat -- there are obviously many polynomials that have no roots whatsoever in the real numbers -- such as x² + 1. However, there is a larger number system, called the complex numbers, which is created by adjoining to the real numbers a new element i with the property that i² = -1 (not a property that any real number can have), and then taking all the elements of the form a+bi, where a and b are real numbers. This obviously allows you to solve any quadratic equation using the quadratic formula, but what the fundamental theorem of algebra says is that this suffices to solve any polynomial whatsoever. Note that this even applies to polynomials with complex coefficients, such as (3+i)x² + 4i. So the fundamental theorem of algebra expresses that the complex numbers are algebraically closed -- adding a solution to just one type of polynomial (namely, x²+1) actually suffices to provide solutions to all of them.

So what to make of all this? The practical upshot of all this is that you can factor any nonzero polynomial whatsoever into the form k(x - r₁)(x - r₂)(x - r₃)... (x - r_n), where k is a nonzero constant and r₁, r₂... r_n are the roots of the polynomial. Further, this factorization is unique, up to the order of the factors. Thus, in essence, the fundamental theorem of algebra is analogous to the fundamental theorem of arithmetic, in that it guarantees the existence of a factorization into suitable "prime" elements. It also means that you can recognize when you've found all the possible solutions to a polynomial equation -- if the polynomial has degree n, then there are n solutions to find, so if you have n (not necessarily distinct, see the above note about multiplicity) solutions, you're done. Conversely, if you don't have n solutions, you're not done.

Answer 2

Carl Friedrich Gauss proved that every equation has exactly as many positive, negative, or complex solutions as the degree of the equation itself. Every first degree equation (linear equations) has one solution, every second-degree equation (quadratic equations) has two solutions, every third-degree equation has three solutions, every "nth" degree equation has n solutions.
This satisfying symmetry is called the "fundamental
theorem of algebra".

By degree of an equation, we mean the highest power found in the equation. Sometimes it's hard to see, when 3 or 4 variables are together. For example
x^2y^3z^4 is degree 9. It's easy to see this if you consider what happens to x^2y^3z^4when x=y=z.
It becomes (x^2)(x^3)(x^4), or x^9. Hence, degree 9
Does this explanation work for you?

Answer 3

There is nothing in algebra. Its some statement like x+y+a, (a+b)^2 +c,etc. I am bad explaining. But its very easy and you will know in your way

Answer 4

What puzzles me is what shape would it take, having ten sides!!! Would you be able to roll it?

Answer 5

im in algebra 1 too. you could try hotmath.com. i would help you but idk what it is you need help on. i find math pretty easy.