## Algebra## Polynomial Equations |

## How do you find the roots of a polynomial? |

Finding the root, also called a zero, of a polynomial is one way to solve for the equation. In other words, the root of an equation is simply a number (or numbers) that solves the equation.

For example, for second degree polynomials, we can find the roots by *completing the square.* Picking apart an equation is the best way to see this:

- 3x
^{2}- 4x + 1 = 0 - (1/3)( 3x
^{2}- 4x + 1) = (1/3)0 (making the coefficient of the x^{2}term into a 1) - x
^{2}- (4/3)x + 1/3 = 0 - (x
^{2}- (4/3)x) + 1/3 = 0 (group the*x*and x^{2}terms together) - (x
^{2}- (4/3)x + (-2/3)^{2}) - (-2/3)^{2}+ 1/3 = 0 (determine the coefficient of the*x*term, divide it by 2 and then square; add and subtract that term) - (x - 2/3)
^{2}- 4/9 + 1/3 = 0 - (x - 2/3)
^{2}- 1/9 = 0 (add together the 4/9 + 1/3 by converting the denominator to 9, in which 1/3 becomes 3/9) - (x - 2/3)
^{2}= 1/9 (move the 1/9 to the other side of the equation by subtracting it from both sides) - x - 2/3 = 1/3 or × - 2/3 = -1/3

That means that x = 1 or x = 1/3 are the two roots that make the equation true (just substitute either number into the initial equation to see that they are both true).