cube roots??

Question

how to find the roots of a polynomial eqn??
say atleast for finding cube roots plz..

dont say that we hav to factorise

i need even the complex root of that

Answer 1

Consider a cubic polynomial equation:

a.x^3+b.x^2+c.x+d=0

Use numerical techniques like :
1) fixed point method
2) Newton ralphson method
3) bisection method
4) false position method
5) chord method
to find one of the roots and then the equation will reduce to a quadratic equation which can be solved even for complex roots

Answer 2

We can find the roots of a polynomial with the following methods,

1.The factor theorem

2. The fundamental theorem of algebra

3. A strategy for finding roots

4. The integer root theorem

5. Conjugate pairs

6. Proof of the factor theorem

7. Proof of the integer root theorem


We will see in that topic what is called the factor theorem.



The Factor Theorem. x − r is a factor of a polynomial P(x) if and only if r is a root of P(x).

This means that if a polynomial can be factored, for example, as follows:

P(x) = (x − 1)(x + 2)(x + 3)

then the theorem tells us that the roots are 1, −2, and −3.

Conversely, if we know that roots of a polynomial are −2, 1, and 5, then the polynomial has the following factors:

(x + 2)(x − 1)(x − 5)

We will see how to prove the factor theorem below.

Sample:

Use the Factor Theorem to prove: (x + 1) is a factor of x5 + 1.

Solution :

−1 is a root of x^5 + 1. For, (−1)^5 + 1 = −1 + 1 = 0.
Therefore, according to the Factor Theorem,
[x −(−1)] = (x + 1) is a factor.

For other methods go through the link

http://www.themathpage.com/aPreCalc/factor-theorem.htm

Answer 3

Cube root on a standard calculator

From the identity:

\frac{1}{3} = \frac{1}{2^2} \left(1 + \frac{1}{2^2}\right) \left(1 + \frac{1}{2^4}\right) \left(1 + \frac{1}{2^8}\right) \left(1 + \frac{1}{2^{16}}\right) \dots,

there is a simple method to compute the cube roots using a non-scientific calculator, which requires only the multiplication and square root buttons. No memory is required. The following method is used:

* Press the square root button once.
* Press the multiplication button.
* Press the square root button twice.
* Press the multiplication button.
* Press the square root button four times.
* Press the multiplication button.
* Press the square root button eight times.
* Press the multiplication button...

This process is continued until the number does not change when the multiplication button is pressed, since the repeated square root gives 1 (this means that the solution has been determined to as many significant digits as the calculator can handle). Then, press the square root button one last time. At this point an approximation of the cube root of the original number will be shown in the display.

If the first multiplication is replaced by division, instead of the cube root, the fifth root will be shown on the display.
[edit]

Why this method works

After raising x to the power on both sides of the above identity:

x^{\frac{1}{3}} = x^{\frac{1}{2^2} \left(1 + \frac{1}{2^2}\right) \left(1 + \frac{1}{2^4}\right) \left(1 + \frac{1}{2^8}\right) \left(1 + \frac{1}{2^{16}}\right) ...} (*)

The left hand side is the cube root of x.

The steps shown in the method give:

After the second step:

x^{\frac{1}{2}}

After the fourth step:

x^{\frac{1}{2} (1 + \frac{1}{2^2})}

After the sixth step:

x^{\frac{1}{2} (1 + \frac{1}{2^2}) (1 + \frac{1}{2^4})}

After the eighth step:

x^{\frac{1}{2} (1 + \frac{1}{2^2}) (1 + \frac{1}{2^4}) (1 + \frac{1}{2^8})}

etc.

Once the value of the expression is equal to 1 to the accuracy of the calculator, the final square root will return the right hand of (*).

Answer 4

You raise it to the 1/3 power on a calculator.

Answer 5

there are formulas for the sum and difference of perfect cubes. There is also a formula similar to the quadratic formula but for cubes. google cube roots, what you need is there.

Answer 6

use horner method to find the roots , for degree 3