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Roots of Polynomial

TopWhen equations are written in axn + bxn-1 + cxn-3 + ….......+z = 0, where a, b, c..... Are constants and are not equals to 0, than this equation is known as a polynomial equation. Polynomial equations are divided into two parts -
For finding roots of polynomial, we use following steps -
Step 1: First of all, we convert our high order polynomial equation into Quadratic Equation by using the division method, like we have a high order polynomial equation,
x3 + px2 + qx + r = 0, then we substitute ‘x’ as a (y – p/3),
It gives a simple polynomial equation y3 + ay + b = 0,
Then we convert this equation into quadratic equation-
(y – v) . (y2 + ay + b) = 0,
Step 2: After evaluation of a quadratic equation, we get one root (y – v) and other root is calculated by following roots of this quadratic equation (y2 + ay + b),
y = [-a +/- √(a2 – 4b)]/2,
This gives two roots of ‘y’.
So, finally we get 3 roots of this cubic polynomial equation.
Now we take an example to understand the process of finding roots of polynomial:
Example: Find the roots of following polynomial equation -
x4 – 2x2 – 3x – 2 = 0?
Solution: We use following steps for evaluating the roots of given polynomial equation -
Step 1: First of all we convert equation into quadratic equation,
x4 – 2x2 – 3x – 2 = 0,
=> (x + 1)(x - 2)(x2 + x + 1) = 0,
Step 2: Now we calculate roots of quadratic equation -
x2 + x + 1 = 0,
x = -1/2 ± √3/2i,
So, finally we get following roots of polynomial -
(x + 1) (x – 2) and x = -1/2 ± √3/2i.