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Solution - Polynomial root calculator

x = - 0.130478

x=-0.130478

x = - 0.778 - 0.115 i

x=-0.778-0.115i

x = - 0.778 + 0.115 i

x=-0.778+0.115i

x = 0.844 - 0.250 i

x=0.844-0.250i

x = 0.844 + 0.250 i

x=0.844+0.250i

See steps Try this:

Step-by-step explanation

1. Normalize to P(x)=0

$16 x^{5} - 20 x^{3} + 8 x + 1 = 0$

$a_{n} = 16$

$a_{0} = 1$

Rewrite in standard form P(x)=0, then identify the leading and constant coefficients.

2. Test candidate roots

Build rational candidates from factor ratios p/q: -1/16, 1/16, -1/8, 1/8, -1/4, 1/4, -1/2, 1/2, -1, 1. Evaluate P(c) for each candidate and deflate when P(c)=0.

$f a c t o r s (| a_{0} |) = 1$

$f a c t o r s (| a_{n} |) = 1, 2, 4, 8, 16$

$x = + / - p / q = - 1 / 16, 1 / 16, - 1 / 8, 1 / 8, - 1 / 4, 1 / 4, - 1 / 2, 1 / 2, - 1, 1$

$P (- 1 / 16) = 0.504868$

$P (1 / 16) = 1.495132$

$P (- 1 / 8) = 0.038574$

$P (1 / 8) = 1.961426$

$P (- 1 / 4) = - 0.703125$

$P (1 / 4) = 2.703125$

$P (- 1 / 2) = - 1$

$P (1 / 2) = 3$

$P (- 1) = - 3$

$P (1) = 5$

3. Solve the remaining factor

After deflation, solve the remaining factor using numeric approximation (Durand-Kerner).

$x = 0.843501 - 0.250127 i, 0.843501 + 0.250127 i, - 0.778263 + 0.114612 i, - 0.778263 - 0.114612 i, - 0.130478$

4. Collect all roots

All roots with multiplicity: x={-0.130478,-0.778263-0.114612i,-0.778263+0.114612i,0.843501-0.250127i,0.843501+0.250127i}.

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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

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