# How do you divide #( 2x^4 - 5x^3 - 8x^2+17x+1 )/(x^2 - 2 )#?

When attempting to divide two rational functions, we should check to see if the denominator and the numerator are zeros first.

#(2 x^4 - 5 x^3 - 8 x^2 + 17 x + 1 )/(x^2 - 2)= (2 x^2 - 5 x - 4) + (7 x - 7 )/(x^2-2)#

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To divide (2x^4 - 5x^3 - 8x^2 + 17x + 1) by (x^2 - 2), you can use polynomial long division.

First, divide the highest degree term of the numerator (2x^4) by the highest degree term of the denominator (x^2). This gives you 2x^2.

Next, multiply the entire denominator (x^2 - 2) by the result (2x^2), which gives you 2x^4 - 4x^2.

Subtract this result from the numerator (2x^4 - 5x^3 - 8x^2 + 17x + 1) to get -x^3 - 4x^2 + 17x + 1.

Now, repeat the process with the new numerator (-x^3 - 4x^2 + 17x + 1) and the denominator (x^2 - 2).

Divide the highest degree term of the new numerator (-x^3) by the highest degree term of the denominator (x^2). This gives you -x.

Multiply the entire denominator (x^2 - 2) by the result (-x), which gives you -x^3 + 2x.

Subtract this result from the new numerator (-x^3 - 4x^2 + 17x + 1) to get -6x^2 + 15x + 1.

Repeat the process with the new numerator (-6x^2 + 15x + 1) and the denominator (x^2 - 2).

Divide the highest degree term of the new numerator (-6x^2) by the highest degree term of the denominator (x^2). This gives you -6.

Multiply the entire denominator (x^2 - 2) by the result (-6), which gives you -6x^2 + 12.

Subtract this result from the new numerator (-6x^2 + 15x + 1) to get 3x + 1.

Since the degree of the new numerator (3x + 1) is less than the degree of the denominator (x^2 - 2), you have reached the end of the division.

Therefore, the quotient is 2x^2 - x - 6 with a remainder of 3x + 1.

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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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