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

Answer 1

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 #0#'s for any missing powers of #x#.

I like to long divide the coefficients, not forgetting to include #0#'s for any missing powers of #x#...

This is similar to long division of numbers.

Reconstructing polynomials from the resulting sequences we find that the quotient is #x^3+x-1# with remainder #-15x+13#

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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Answer 2

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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Answer from HIX Tutor

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.

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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