How to Do Long Division with Polynomials
Read the problem., Set up a long division problem., Estimate the first term of your quotient., Multiply your first term by the divisor., Subtract., Carry down the next term of the dividend., Start the process over again., Multiply the last term of...
Step-by-Step Guide
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Step 1: Read the problem.
The problem may be presented to you as a straightforward division problem, with instructions to find the quotient.
You may also have a fraction, with one polynomial as the numerator and a binomial as the denominator.
You should recognize this as an opportunity to perform division.For example, a division problem could be stated as, “Find the quotient when 3x2+20x+12{\displaystyle 3x^{2}+20x+12} is divided by x+6{\displaystyle x+6}.” The same problem could ask you, “One factor of 3x2+20x+12{\displaystyle 3x^{2}+20x+12} is x+6{\displaystyle x+6}.
What is the other factor?” Finally, the exact same problem may just appear as 3x2+20x+12x+6{\displaystyle {\frac {3x^{2}+20x+12}{x+6}}}.
You should recognize that the fraction form means to divide the numerator by the denominator. -
Step 2: Set up a long division problem.
Just as you would with numbers, begin by drawing a long division symbol, something like this: )¯¯¯¯¯¯ .
The polynomial that is your dividend goes in the space under the symbol.
The divisor is placed to the left of the symbol.The "dividend" is the large term whose factors you are trying to find.
The "divisor" is the factor that you are dividing by.
The "quotient" is the answer of any division problem.
With polynomials, this problem will look like: x+6)3x2+20x+12¯{\displaystyle x+6{\overline {)3x^{2}+20x+12}}}. , When you are doing long division with numbers, you don’t try to divide the entire number in one step.
You look at the first one or two numbers of the dividend and estimate how many times the first digit of the divisor will go into that.
You will do the same with polynomial division.
Look at the first term of the divisor and decide how many times that will go into the first term of the dividend.For example, if you are dividing 642 by 3, you begin by considering how many times 3 will divide into the first digit of
642.
Three goes into six twice, so you will write a 2 above the 6 over the division line.
For the polynomial division, consider the first term of the dividend, 3x2{\displaystyle 3x^{2}} and the first term of the divisor, x{\displaystyle x}. 3x2{\displaystyle 3x^{2}} divided by x{\displaystyle x} leaves a factor of 3x{\displaystyle 3x}.
Write 3x{\displaystyle 3x} above the 3x2{\displaystyle 3x^{2}} under the division symbol. , With the first time of your quotient set above the bar line, now multiply that by the full divisor.
Write the result underneath the dividend.With 3x{\displaystyle 3x} as the first term of your quotient, multiply 3x{\displaystyle 3x} by x+6{\displaystyle x+6}.
Do this by multiplying 3x by each term.
First do 3x∗x{\displaystyle 3x*x} and then 3x∗+6{\displaystyle 3x*+6}.
Write the result, 3x2+18x{\displaystyle 3x^{2}+18x} underneath the first two terms of the polynomial 3x2+20x{\displaystyle 3x^{2}+20x}. , Just as the next step in long division is to subtract your result from the original number, in this problem you will subtract the polynomial minus the binomial you just wrote down.
You should have written your previous step underneath similar terms of the polynomial, so you can simply subtract downward.
Draw a line underneath the lower binomial and subtract.In the running example, the first terms should line up to subtract 3x2−3x2{\displaystyle 3x^{2}-3x^{2}}.
This cancels out to zero.
Then subtract the second terms, 20x−18x{\displaystyle 20x-18x}.
Below the subtraction line, write your answer of 2x{\displaystyle 2x}. , In numerical long division, you would now bring down the next digit of the number.
In polynomial long division, copy down the next term of the polynomial.In this example, the next (and last) term of the polynomial is +12{\displaystyle +12}.
Copy that down to the bottom, next to the 2x{\displaystyle 2x}, to create the binomial 2x+12{\displaystyle 2x+12}. , Compare this new dividend, 2x+2{\displaystyle 2x+2} to the divisor x+6{\displaystyle x+6}.
Consider how many times the first term, 2x{\displaystyle 2x} can divide the first term of the divisor x{\displaystyle x}. 2x{\displaystyle 2x} divided by x{\displaystyle x} is {\displaystyle }.
Write this result, {\displaystyle } as the next term of your quotient at the top of the problem.Because the {\displaystyle } is positive, write it as +2{\displaystyle +2}.
This will give the quotient of 3x+2{\displaystyle 3x+2} above the division line. , Continue the process by multiplying.In this example, multiply the +2{\displaystyle +2} times each term of the divisor x+6{\displaystyle x+6}.
This will give the result 2x+12{\displaystyle 2x+12}.
Write this result at the bottom of the long division problem, lining up the terms with the result of your prior subtraction. , Line up common terms and then subtract.
The binomial at the bottom of the problem from your prior subtraction was 2x+12{\displaystyle 2x+12}.
Underneath that is the latest product, which is also 2x+12{\displaystyle 2x+12}.
When you subtract each term, the result will be zero., When you have used all the terms of the initial polynomial, and your subtraction cancels out all terms to zero, you are done with the long division.
The result of 3x2+20x+12{\displaystyle 3x^{2}+20x+12} divided by x+6{\displaystyle x+6} is 3x+2{\displaystyle 3x+2}.Alternatively, if working with the problem in fraction form, the result will look like this: 3x2+20x+12x+6=(3x+2)(x+6)x+6=3x+2{\displaystyle {\frac {3x^{2}+20x+12}{x+6}}={\frac {(3x+2)(x+6)}{x+6}}=3x+2} -
Step 3: Estimate the first term of your quotient.
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Step 4: Multiply your first term by the divisor.
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Step 5: Subtract.
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Step 6: Carry down the next term of the dividend.
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Step 7: Start the process over again.
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Step 8: Multiply the last term of the quotient by the divisor.
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Step 9: Subtract.
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Step 10: Report your result.
Detailed Guide
The problem may be presented to you as a straightforward division problem, with instructions to find the quotient.
You may also have a fraction, with one polynomial as the numerator and a binomial as the denominator.
You should recognize this as an opportunity to perform division.For example, a division problem could be stated as, “Find the quotient when 3x2+20x+12{\displaystyle 3x^{2}+20x+12} is divided by x+6{\displaystyle x+6}.” The same problem could ask you, “One factor of 3x2+20x+12{\displaystyle 3x^{2}+20x+12} is x+6{\displaystyle x+6}.
What is the other factor?” Finally, the exact same problem may just appear as 3x2+20x+12x+6{\displaystyle {\frac {3x^{2}+20x+12}{x+6}}}.
You should recognize that the fraction form means to divide the numerator by the denominator.
Just as you would with numbers, begin by drawing a long division symbol, something like this: )¯¯¯¯¯¯ .
The polynomial that is your dividend goes in the space under the symbol.
The divisor is placed to the left of the symbol.The "dividend" is the large term whose factors you are trying to find.
The "divisor" is the factor that you are dividing by.
The "quotient" is the answer of any division problem.
With polynomials, this problem will look like: x+6)3x2+20x+12¯{\displaystyle x+6{\overline {)3x^{2}+20x+12}}}. , When you are doing long division with numbers, you don’t try to divide the entire number in one step.
You look at the first one or two numbers of the dividend and estimate how many times the first digit of the divisor will go into that.
You will do the same with polynomial division.
Look at the first term of the divisor and decide how many times that will go into the first term of the dividend.For example, if you are dividing 642 by 3, you begin by considering how many times 3 will divide into the first digit of
642.
Three goes into six twice, so you will write a 2 above the 6 over the division line.
For the polynomial division, consider the first term of the dividend, 3x2{\displaystyle 3x^{2}} and the first term of the divisor, x{\displaystyle x}. 3x2{\displaystyle 3x^{2}} divided by x{\displaystyle x} leaves a factor of 3x{\displaystyle 3x}.
Write 3x{\displaystyle 3x} above the 3x2{\displaystyle 3x^{2}} under the division symbol. , With the first time of your quotient set above the bar line, now multiply that by the full divisor.
Write the result underneath the dividend.With 3x{\displaystyle 3x} as the first term of your quotient, multiply 3x{\displaystyle 3x} by x+6{\displaystyle x+6}.
Do this by multiplying 3x by each term.
First do 3x∗x{\displaystyle 3x*x} and then 3x∗+6{\displaystyle 3x*+6}.
Write the result, 3x2+18x{\displaystyle 3x^{2}+18x} underneath the first two terms of the polynomial 3x2+20x{\displaystyle 3x^{2}+20x}. , Just as the next step in long division is to subtract your result from the original number, in this problem you will subtract the polynomial minus the binomial you just wrote down.
You should have written your previous step underneath similar terms of the polynomial, so you can simply subtract downward.
Draw a line underneath the lower binomial and subtract.In the running example, the first terms should line up to subtract 3x2−3x2{\displaystyle 3x^{2}-3x^{2}}.
This cancels out to zero.
Then subtract the second terms, 20x−18x{\displaystyle 20x-18x}.
Below the subtraction line, write your answer of 2x{\displaystyle 2x}. , In numerical long division, you would now bring down the next digit of the number.
In polynomial long division, copy down the next term of the polynomial.In this example, the next (and last) term of the polynomial is +12{\displaystyle +12}.
Copy that down to the bottom, next to the 2x{\displaystyle 2x}, to create the binomial 2x+12{\displaystyle 2x+12}. , Compare this new dividend, 2x+2{\displaystyle 2x+2} to the divisor x+6{\displaystyle x+6}.
Consider how many times the first term, 2x{\displaystyle 2x} can divide the first term of the divisor x{\displaystyle x}. 2x{\displaystyle 2x} divided by x{\displaystyle x} is {\displaystyle }.
Write this result, {\displaystyle } as the next term of your quotient at the top of the problem.Because the {\displaystyle } is positive, write it as +2{\displaystyle +2}.
This will give the quotient of 3x+2{\displaystyle 3x+2} above the division line. , Continue the process by multiplying.In this example, multiply the +2{\displaystyle +2} times each term of the divisor x+6{\displaystyle x+6}.
This will give the result 2x+12{\displaystyle 2x+12}.
Write this result at the bottom of the long division problem, lining up the terms with the result of your prior subtraction. , Line up common terms and then subtract.
The binomial at the bottom of the problem from your prior subtraction was 2x+12{\displaystyle 2x+12}.
Underneath that is the latest product, which is also 2x+12{\displaystyle 2x+12}.
When you subtract each term, the result will be zero., When you have used all the terms of the initial polynomial, and your subtraction cancels out all terms to zero, you are done with the long division.
The result of 3x2+20x+12{\displaystyle 3x^{2}+20x+12} divided by x+6{\displaystyle x+6} is 3x+2{\displaystyle 3x+2}.Alternatively, if working with the problem in fraction form, the result will look like this: 3x2+20x+12x+6=(3x+2)(x+6)x+6=3x+2{\displaystyle {\frac {3x^{2}+20x+12}{x+6}}={\frac {(3x+2)(x+6)}{x+6}}=3x+2}
About the Author
Danielle Ryan
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