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.
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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.
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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.
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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!
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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.
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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, −3x3 + 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
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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:
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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.
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So the final answer is
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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.
<figure></figure>STEP 6: Perform subtraction by alternating the signs of the bottom polynomial.
<figure></figure>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
<figure></figure>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.
<figure></figure>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!
<figure></figure>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!
<figure></figure>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”.
<figure></figure>STEP 16: Finish this off by subtraction leaving as with a remainder of −7.
<figure></figure>STEP 17: Write the final answer in the following form.
<figure></figure> <figure></figure> <hr />Example 3: Divide using the long division method
<figure></figure>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.
<figure></figure>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.
<figure></figure>STEP 7: Carry down the next unused term of the dividend.
<figure></figure>STEP 8: Looking at the bottom polynomial, −3x3 + 0x2, use the leading term −3x3 and divide it by the leading term of the divisor, x. Put the answer above the division symbol.
<figure></figure>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.
<figure></figure>STEP 11: Bring down the next adjacent term of the dividend
<figure></figure>STEP 12: Go up again by dividing the leading term below by the leading term of the divisor.
<figure></figure>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?
<figure></figure>STEP 14: Carry down the last term of the dividend.
<figure></figure>STEP 15: Go up again while performing division.
<figure></figure>STEP 16: Go down again while performing multiplication.
<figure></figure>STEP 17: Do the final subtraction, and we are done! The remainder is equal to 20.
<figure></figure>STEP 18: The final answer is
<figure></figure> <figure></figure> <hr />Example 4: Divide the given polynomial using long division method
<figure></figure>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:
Verify if the steps are being applied correctly in the example below.
<figure></figure>So the final answer is
<figure></figure>0Thank You