# How do you find all the real and complex roots of # x^6 - 4x^5 - 24x^2 + 10x - 3 = 0#?

Use Newton's method and a spreadsheet to find approximations:

#-1.7480395639#

#4.2786545820#

#0.2098743850+-0.2854969683i#

#0.5248181059+-1.7085773197i#

We can find approximations for the roots using Newton's method.

We can use this with a spreadsheet to find good approximations to the roots in a few iterations.

If your spreadsheet application is like the one I use, it does not handle Complex numbers directly, so to find the Complex roots we need to have separate columns for Real and imaginary parts.

I used formulas for the first row like the following:

With these definitions:

Then the second row is similar to the first except for the first two columns:

Copy the second row a few times (allowing the spreadsheet application to auto increment the row references)

Hence find the following roots:

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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