How do you divide #( 6x^3 + 10x^2 + x + 8)/(2x^2 + 1)#?

Answer 1

Long divide the coefficients to find quotient #3x+5# with remainder #-2x+3#.

I like to long divide the coefficients like this:

...not forgetting to include a #0# in the divisor to stand for the missing #x# term.

In this example we find the quotient is #3x+5# with remainder #-2x+3#

The process is similar to long division of numbers:

Write the dividend (#6, 10, 1, 8#) under the bar.

Write the divisor (#2, 0, 1#) to the left of the bar.

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 #color(blue)(3)#, so that when multiplied by #2, 0, 1#, the resulting first term matches the leading #6# of the dividend.

We then write out the product #6, 0, 3# of #3# and #2, 0, 1# under the dividend and subtract it to give #10, -2#. We bring down the next term from the dividend alongside it to give our running remainder.

We choose the next term of the quotient #color(blue)(5)#, so that when multiplied by #2, 0, 1#, the resulting first term matches the leading term #10# of our running remainder.

We then write out the produce #10, 0, 5# of #5# and #2, 0, 1# under the running remainder and subtract it to give #-2, 3#.

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 #x# to find:

#(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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Answer 2

To divide (6x^3 + 10x^2 + x + 8) by (2x^2 + 1), you can use long division. Here are the steps:

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

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

  3. 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.

  4. Bring down the next term from the numerator (10x^2) and repeat steps 1-3.

  5. 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.

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

  7. 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.

  8. Bring down the next term from the numerator (x) and repeat steps 1-3.

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

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

  11. 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.

  12. Bring down the last term from the numerator (8) and repeat steps 1-3.

  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.

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

  15. 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.

  16. 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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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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