# How do you divide #( x^5 - x^3 - x^2 - 17x - 15 )/(x^2 - 2 )#?

Long divide the coefficients to find:

#(x^5-x^3-x^2-17x-15)/(x^2-2) = (x^3+x-1) + (-15x+13)/(x^2-2)#

...not forgetting to include

I like to long divide the coefficients, not forgetting to include

This is similar to long division of numbers.

Reconstructing polynomials from the resulting sequences we find that the quotient is

That is:

#(x^5-x^3-x^2-17x-15)/(x^2-2) = (x^3+x-1) + (-15x+13)/(x^2-2)#

Or if you prefer:

#(x^5-x^3-x^2-17x-15) = (x^2-2)(x^3+x-1) + (-15x+13)#

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To divide (x^5 - x^3 - x^2 - 17x - 15) by (x^2 - 2), you can use polynomial long division.

First, divide the highest degree term of the numerator (x^5) by the highest degree term of the denominator (x^2). This gives you x^3.

Next, multiply the entire denominator (x^2 - 2) by the result (x^3), which gives you x^5 - 2x^3. Subtract this from the numerator (x^5 - x^3 - x^2 - 17x - 15) to get -x^3 - x^2 - 17x - 15 + (x^5 - 2x^3). Simplify this to x^5 - 3x^3 - x^2 - 17x - 15.

Repeat the process by dividing the highest degree term of the new numerator (x^5) by the highest degree term of the denominator (x^2), which gives you x^3. Multiply the entire denominator (x^2 - 2) by x^3, which gives you x^5 - 2x^3. Subtract this from the new numerator (x^5 - 3x^3 - x^2 - 17x - 15) to get -x^3 - x^2 - 17x - 15 + (x^5 - 2x^3). Simplify this to -x^3 - x^2 - 17x - 15.

Repeat this process until the degree of the new numerator is less than the degree of the denominator. In this case, the final result is x^3 - x - 8, with a remainder of -7x - 7.

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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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