How do you divide #( 6x^3 + 10x^2 + x + 8)/(2x^2 + 1)#?
Long divide the coefficients to find quotient
I like to long divide the coefficients like this:
...not forgetting to include a
In this example we find the quotient is
The process is similar to long division of numbers:
Write the dividend (
Write the divisor (
Start writing the quotient, term by term, choosing each successive term to match the leading term of the running remainder:
The first term of the quotient is
We then write out the product
We choose the next term of the quotient
We then write out the produce
There are no more terms to bring down from the dividend, so this is our final remainder.
Then interpret the coefficient sequences by applying them to the appropriate powers of
#(6x^3+10x^2+x+8)/(2x^2+1) = 3x+5+(2x+3)/(2x^2+1)#
Or if you prefer:
#6x^3+10x^2+x+8 = (3x+5)(2x^2+1)+(2x+3)#
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To divide (6x^3 + 10x^2 + x + 8) by (2x^2 + 1), you can use long division. Here are the steps:

Divide the highest degree term of the numerator (6x^3) by the highest degree term of the denominator (2x^2). This gives you 3x.

Multiply the entire denominator (2x^2 + 1) by the quotient obtained in step 1 (3x). This gives you 6x^3 + 3x.

Subtract the result obtained in step 2 from the numerator (6x^3 + 10x^2 + x + 8) to get the remainder. The remainder is 10x^2  3x + 8.

Bring down the next term from the numerator (10x^2) and repeat steps 13.

Divide the new highest degree term of the numerator (10x^2) by the highest degree term of the denominator (2x^2). This gives you 5.

Multiply the entire denominator (2x^2 + 1) by the quotient obtained in step 5 (5). This gives you 10x^2 + 5.

Subtract the result obtained in step 6 from the remainder obtained in step 3 (10x^2  3x + 8) to get a new remainder. The new remainder is 3x + 3.

Bring down the next term from the numerator (x) and repeat steps 13.

Divide the new highest degree term of the numerator (x) by the highest degree term of the denominator (2x^2). This gives you 0.

Multiply the entire denominator (2x^2 + 1) by the quotient obtained in step 9 (0). This gives you 0.

Subtract the result obtained in step 10 from the remainder obtained in step 8 (0 + 3) to get a new remainder. The new remainder is 3.

Bring down the last term from the numerator (8) and repeat steps 13.

Divide the new highest degree term of the numerator (8) by the highest degree term of the denominator (2x^2). This gives you 4.

Multiply the entire denominator (2x^2 + 1) by the quotient obtained in step 13 (4). This gives you 8x^2 + 4.

Subtract the result obtained in step 14 from the remainder obtained in step 12 (8  4) to get a new remainder. The new remainder is 4.

Since the degree of the new remainder (4) is less than the degree of the denominator (2x^2 + 1), you have finished the division.
The final result of the division is the quotient obtained from the steps: 3x + 5. The remainder is 4.
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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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