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Fundamental Theorem of Algebra

Reference > Mathematics > Algebra > Polynomials
 

A zero of a polynomial is a value of the  variable for which the polynomial equals zero.

For example, given the polynomial 2x - 4, it is easy to see that when x = 2, the polynomial is equal to zero. Thus, we say that 2 is a zero of the polynomial. In the case of this polynomial, it was easy to find the zero by inspection, but it won't always be that easy, so we should have an algebraic way of finding the zeroes.

The algebraic way, it turns out, is fairly straightforward, and you may have already thought of it: you simply set the polynomial equal to zero, and solve:

2x - 4 = 0
2x = 4
x = 2.

When you set a polynomial equal to zero (or any other value) it is no longer a polynomial, but a polynomial equation.

Example: Find the zeroes of the polynomial x+ 3x + 2

Solution: Set the polynomial equal to 0.
x+ 3x + 2 = 0
(x + 2)(x + 1) = 0
x = -1 or x = -2

Thus, this polynomial has two zeroes: -1 and -2

Example: Find the zeroes of the polynomial x3 - 4x2 - 4x + 16

Solution: Set the polynomial equal to 0, then group the terms in pairs to factor.
x3 - 4x2 - 4x + 16 = 0
(x3 - 4x2) - (4x - 16) = 0
x2(x - 4) - 4(x - 4) = 0
(x2 - 4)(x - 4) = 0
(x - 2)(x + 2)(x - 4) = 0
x = 2, x = -2, or x = 4

Example: Find the values of x for which the polynomial x2 + 5x is equal to 6.

Solution: Here we set the polynomial equal to 6 instead of zero.
x2 + 5x = 6
x2 + 5x - 6 = 0
(x - 1)(x + 6) = 0
x = 1 or x = -6

It's interesting, isn't it, that our first degree polynomial had one zero, our second degree polynomial had two zeroes, and our third degree polynomial had three zeroes. Do you suppose that's a consistent pattern? Well, yes it is (with a couple stipulations). The idea that an nth degree polynomial has n roots is called the Fundamental Theorem of Algebra.

Fundamental Theorem of Algebra
Every non-zero, single variable polynomial of degree n has exactly n zeroes, with the following caveats:

  • If, algebraically, we find the same zero k times, we count it as k separate zeroes.
  • Some of the roots may be non-Reals (another way of saying this: the zeroes lie in the complex plane).

Example: The polynomial x2 - 2x + 1 has only one zero (at x = 1). However, when solved algebraically, we obtain (x - 1)2 = 0, so x = 1 counts twice, leading to the total of 2 roots for a second degree polynomial.

Example: The polynomial x2 + 2x + 2 has no real roots. However, if we use the Quadratic Formula on it, we find that it has two complex roots: 1 + i and 1 - i.

Questions

1.
How many roots does the following equation have? 2x + 3x3 - 1 = 0
2.
How many roots does the following equation have? (2x2 + 2x -1)(3x2 - 9) = 0
3.
How many roots does the equation (x + 1)3 = 0 have? How many distinct roots does it have?
4.
Why does the Fundamental Theorem of Algebra specify "non-zero"?
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