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Long division method 5examples

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Long division method 5examples
  • 1 answers

Gaurav Seth 3 years, 3 months ago

STEP 1: Focus on the leftmost terms of both the dividend and divisor.

STEP 2: Divide the leftmost term of the dividend by the leftmost term of the divisor.

STEP 3: Place the partial answer on top.

<figure></figure>

STEP 4: Use that partial answer, x2, to multiply into the divisor (3x−2).

STEP 5: Place their product under the dividend. Make sure to align them by similar terms.

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STEP 6: Perform subtraction by alternating the signs of the bottom polynomial.

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STEP 7: Proceed with regular addition vertically. Again the first column cancels each other out. Looks like a pattern to me!

STEP 8: Carry down the next adjacent “unused” term of the dividend

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STEP 9: Take the leftmost term of the bottom polynomial, and divide by the leftmost term of the divisor.

STEP 10: Place the answer on top, as usual.

<figure></figure>

STEP 11: Okay, perform another multiplication by the partial answer 2x and divisor (3x−2). Bring the product below.

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STEP 12: Perform subtraction by switching signs and proceed with normal addition.

STEP 13: Carry down the last unused term of the dividend. We’re almost there!

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STEP 14: We are going up one more time. Divide the leading term of the bottom polynomial by the leading term of the divisor.  Place the answer up there!

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STEP 15: This is our “last trip” going down so we distribute the partial answer −1 by the divisor (3x−2), and placing the product “downstairs”.

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STEP 16: Finish this off by subtraction leaving as with a remainder of −7.

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STEP 17: Write the final answer in the following form.

<figure></figure> <figure></figure> <hr />

Example 3: Divide using the long division method

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Solution:  If you observe the dividend, it is missing some powers of variable x which are x3 and x2. I need to insert zero coefficients as placeholders for missing powers of the variable. This is a critical part to correctly apply the procedures in long division.

So I rewrite the original problem as . Now all x‘s are accounted for!

<figure></figure>

STEP 1: Focus on the leading terms inside and outside the division symbol.

STEP 2: Divide the first term of the dividend by the first term of the divisor.

STEP 3: Position the partial answer on top.

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STEP 4: Use that partial answer placed on top, 3x2 to distribute into the divisor (x + 1).

STEP 5: Put the result under the dividend. Make sure to align them by similar terms.

<figure></figure>

STEP 6: Subtract them together by making sure to switch the signs of the bottom terms before adding.

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STEP 7: Carry down the next unused term of the dividend.

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STEP 8: Looking at the bottom polynomial, −3x+ 0x2, use the leading term −3x3 and divide it by the leading term of the divisor, x. Put the answer above the division symbol.

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STEP 9: Multiply the answer you got previously, −3x3, and distribute into the divisor (x + 1).

STEP 10: Place the answer below then perform subtraction.

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STEP 11: Bring down the next adjacent term of the dividend

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STEP 12: Go up again by dividing the leading term below by the leading term of the divisor.

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STEP 13: Go down by distributing the answer in partial quotient into the divisor, followed by subtraction.

I believe the pattern makes sense now. Yes?

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STEP 14: Carry down the last term of the dividend.

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STEP 15: Go up again while performing division.

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STEP 16: Go down again while performing multiplication.

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STEP 17: Do the final subtraction, and we are done! The remainder is equal to 20.

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STEP 18: The final answer is

<figure></figure> <figure></figure> <hr />

Example 4: Divide the given polynomial using long division method

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Solution: The dividend is obviously missing a lot of variable x. That means I need to insert zero coefficients in every missing power of the variable.

I need to rewrite the problem this way to include all exponents of x in descending order:

<figure></figure>

Remember the Main Steps in Long Division:

  • When going up, we divide
  • When going down, we distribute
  • Subtract
  • Carry down
  • Repeat the process until done

Verify if the steps are being applied correctly in the example below.

<figure></figure>

So the final answer is

<figure></figure>
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