# How do you find all the zeros of #f(x) = -7x^9+x^5-x^2+6 #?

Use Newton's method with suitable first approximations to find approximations of the zeros numerically.

We can use these ninth roots as first approximations for Newton's method:

If your spreadsheet application is anything like mine, it does not handle Complex numbers directly, so expressing this formula requires separate columns for Real and imaginary parts.

I won't bother with that at this time, but I can at least find the Real zero using Real arithmetic:

So it converges quite fast and the first approximation was close to the result.

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To find the zeros of the function ( f(x) = -7x^9 + x^5 - x^2 + 6 ), we set the function equal to zero and solve for ( x ):

[ -7x^9 + x^5 - x^2 + 6 = 0 ]

Unfortunately, there's no general formula to solve polynomial equations of degree higher than 4. However, we can use numerical methods or approximation techniques to estimate the roots. These methods include graphical analysis, Newton's method, or numerical approximation algorithms like the bisection method or Newton-Raphson method.

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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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