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Solution - Polynomial root calculator

x = 2.005406

x=2.005406

x = 9.973044

x=9.973044

x = - 4.987 - 8.637 i

x=-4.987-8.637i

x = - 4.987 + 8.637 i

x=-4.987+8.637i

x = - 1.003 - 1.737 i

x=-1.003-1.737i

x = - 1.003 + 1.737 i

x=-1.003+1.737i

See steps Try this:

Step-by-step explanation

1. Normalize to P(x)=0

$x^{6} - 1000 x^{3} + 8000 = 0$

$a_{n} = 1$

$a_{0} = 8000$

Rewrite the equation as a polynomial equal to zero.

2. Test candidate roots

Possible rational roots: -1, 1, -2, 2, -4, 4, -5, 5, -8, 8, -10, 10, -16, 16, -20, 20, -25, 25, -32, 32, -40, 40, -50, 50, -64, 64, -80, 80, -100, 100, -125, 125, -160, 160, -200, 200, -250, 250, -320, 320, -400, 400, -500, 500, -800, 800, -1000, 1000, -1600, 1600, -2000, 2000, -4000, 4000, -8000, 8000. Test each candidate and deflate when a root is found.

$f a c t o r s (| a_{0} |) = 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 250, 320, 400, 500, 800, 1000, 1600, 2000, 4000, 8000$

$f a c t o r s (| a_{n} |) = 1$

$x = + / - p / q = - 1, 1, - 2, 2, - 4, 4, - 5, 5, - 8, 8, - 10, 10, - 16, 16, - 20, 20, - 25, 25, - 32, 32, - 40, 40, - 50, 50, - 64, 64, - 80, 80, - 100, 100, - 125, 125, - 160, 160, - 200, 200, - 250, 250, - 320, 320, - 400, 400, - 500, 500, - 800, 800, - 1000, 1000, - 1600, 1600, - 2000, 2000, - 4000, 4000, - 8000, 8000$

$P (- 1) = 9001$

$P (1) = 7001$

$P (- 2) = 16064$

$P (2) = 64$

$P (- 4) = 76096$

$P (4) = - 51904$

$P (- 5) = 148625$

$P (5) = - 101375$

$P (- 8) = 782144$

$P (8) = - 241856$

3. Solve the remaining factor

Solve the remaining polynomial using numeric approximation (Durand-Kerner).

$x = - 4.986522 + 8.636909 i, - 4.986522 - 8.636909 i, - 1.002703 - 1.736732 i, - 1.002703 + 1.736732 i, 9.973044, 2.005406$

4. Collect all roots

All roots with multiplicity: x={2.005406,9.973044,-4.986522-8.636909i,-4.986522+8.636909i,-1.002703-1.736732i,-1.002703+1.736732i}.

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Terms and topics

Polynomial root calculator

Solution - Polynomial root calculator

Step-by-step explanation

1. Normalize to P(x)=0

2. Test candidate roots

3. Solve the remaining factor

4. Collect all roots

Why learn this

Terms and topics

Related links

Solution - Polynomial root calculator

Step-by-step explanation

1. Normalize to P(x)=0

2. Test candidate roots

3. Solve the remaining factor

4. Collect all roots

Why learn this

Terms and topics

Related links

Latest Related Drills Solved