How do you divide #( x^3 + 8x^2 + 19x + 12 )/(x^2+2)#?
I like to long divide just the coefficients, not forgetting to include
The process is similar to long division of numbers.
Write the dividend under the bar and the divisor to the left of the bar.
Write out the quotient one term at a time, choosing it to cause the leading terms of the running remainder to match.
Subtract the product of the divisor and the term of the quotient from the running remainder and bring down the next term from the dividend alongside it.
Repeat until the running remainder is shorter than the divisor.
In our case we get a quotient
#(x^3+8x^2+19x+12)/(x^2+2) = x+8 + (17x4)/(x^2+2)#
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To divide (x^3 + 8x^2 + 19x + 12) by (x^2 + 2), you can use long division. Here are the steps:

Divide the first term of the numerator (x^3) by the first term of the denominator (x^2). This gives x as the first term of the quotient.

Multiply the entire denominator (x^2 + 2) by x, and subtract the result from the numerator (x^3 + 8x^2 + 19x + 12). This gives you a new polynomial to work with: (6x^2 + 19x + 12).

Repeat the process by dividing the first term of the new polynomial (6x^2) by the first term of the denominator (x^2). This gives 6x as the next term of the quotient.

Multiply the entire denominator (x^2 + 2) by 6x, and subtract the result from the new polynomial (6x^2 + 19x + 12). This gives you a new polynomial to work with: (7x + 12).

Divide the first term of the new polynomial (7x) by the first term of the denominator (x^2). This gives 0 as the next term of the quotient.

Multiply the entire denominator (x^2 + 2) by 0, and subtract the result from the new polynomial (7x + 12). This gives you a remainder of (7x + 12).
Therefore, the quotient is x + 6x + 0, or simply x + 6. The remainder is 7x + 12.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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