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# 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:

1. 3x2- 4x + 1 = 0
2. (1/3)( 3x2 - 4x + 1) = (1/3)0 (making the coefficient of the x2 term into a 1)
3. x2 - (4/3)x + 1/3 = 0
4. (x2 - (4/3)x) + 1/3 = 0 (group the x and x2 terms together)
5. (x2 - (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)
6. (x - 2/3) 2- 4/9 + 1/3 = 0
7. (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)
8. (x - 2/3) 2 = 1/9 (move the 1/9 to the other side of the equation by subtracting it from both sides)
9. 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).

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