Roots

A root of a polynomial is a solution to the equation where the polynomial is set equal to zero.

The fundamental theorem of algebra states that for a polynomial in one variable, the number of roots is equal to the degree of the polynomial (though some may be double or multiple roots).

Example 1:

Find the roots of the polynomial $x^{2} - 5 x + 6$ .

Equate the polynomial to zero.

$x^{2} - 5 x + 6 = 0$

In this case, the polynomial can be easily factored :

$(x - 2) (x - 3) = 0$

By the zero product property , either $x = 2$ or $x = 3$ .

(This polynomial has degree $2$ , so we have found the $2$ roots.)

In the above example, both roots are positive integers. In other polynomials, roots may involve radicals and/or complex numbers .

Example 2:

Find the roots of the polynomial $2 x^{3} + 2 x^{2} + 3 x$ .

Notice that we can immediately factor out an $x$ .

$x (2 x^{2} + 2 x + 3) = 0$

By the zero product property , either $x = 0$ or $2 x^{2} + 2 x + 3 = 0$ .

So, one root is $0$ . To find the other two roots, we use the quadratic formula :

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Here $a = 2, b = 2, and c = 3$ .

$x = \frac{- 2 \pm \sqrt{2^{2} - 4 (2) (3)}}{2 (2)}$

Simplify.

$x = \frac{- 2 \pm \sqrt{- 20}}{4}$

$x = \frac{- 2 \pm 2 i \sqrt{5}}{4}$

$x = - \frac{1}{2} \pm i \frac{\sqrt{5}}{2}$

So the polynomial has $1$ real root and $2$ complex roots, for a total of $3$ .

Roots

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