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Related Lessons: Adding and Subtracting Polynomials Multiplying Polynomials Synthetic Division Method

Polynomial Long Division | Step by Step

 

In this lesson, I will go over five (5) examples with detailed step by step solution on how to divide polynomials using the long method. It is very similar on what you did back in elementary when you try to divide large numbers, for instance, you have 1723 ÷ 5 = ? You would solve it just like below, right?

long method division with dividend=1723, divisor=5, quotient=344 and 3 as the remainder

If you can do the simple numerical division by long method as shown above, I am convinced that you can do the problems below. The only added thing is division of variables.

Direction: Divide the polynomials using Long Method

Example 1 (detailed solution): (6x^2-2x-28)/(2x+4)

Example 2 (detailed solution): (3x^3+4x^2-7x-5)/(3x-2)

Example 3 (detailed solution): (3x^4-12x+5)/(x+1)

Example 4 (short solution): (-5x^5-1)/(5x+5)

Example 5 (animated solution): (x^4-3x^3+2x^2-x+1)/(x^2-x+1)


Example 1: Divide using long division method (6x^2-2x-28)/(2x+4)

Solution: I need to make sure that both the dividend (stuff being divided) and the divisor are in the standard form. A polynomial in standard form guarantees that their exponents are in decreasing order from left to right. Performing a quick check on this helps us to prevent basic avoidable mistakes later on.

By quick examination, I hope you agree that both our dividend and divisor are indeed in standard form. That means we are now ready to perform the procedure.

Description of step Step by step solution by Long Method

1) Consider both the leading terms of the dividend and divisor.

2) Divide the leading term of the dividend by the leading term of the divisor.

3) Place the partial quotient on top.

 

divide 6x^2 by (2x+4), place 3x on top
   

4) Now take the partial quotient you placed on top, 3x, and distribute into the divisor (2x + 4).

5) Position the product of (3x) and (2x+4) under the dividend.

  • Make sure to align them by similar terms.
distribute 3x to the binomial 2x+4, get 6x^2+12x
   
6) Perform subtraction by switching the signs of the bottom polynomial. switch the signs; so now we have -6x^2-12x
   

7) Proceed with regular addition vertically.

  • Notice that the first column from the left cancels each other out. Nice!
-14x
   
8) Carry down the next adjacent "unused" term of the dividend. -14x-28
   

9) Next, look at the bottom polynomial, −14x−28, take its leading term which is −14x and divide it by the leading term of the divisor, 2x.

10) Again, place the partial quotient on top.

place -7 on top, now we have 3x-7
   

11) Use the partial quotient that you put up, −7, and distribute into the divisor. Seeing a pattern now?

12) Place the product of −7 and the divisor below as the last line of polynomial entry.

multiply -7 to the divisor 2x+4 to get -14x-28
   
13)  Subtraction means you will switch the signs (in red). switch the signs of the polynomial below
   
14) Perform regular addition along the columns of similar terms subtract to get a remainder of zero
   

15) This is great because the remainder is zero. It means the divisor is a factor of the dividend. 

The final answer is just the stuff on top of the division symbol.

(6x^2-2x-28)/(2x+4)=3x-7

Example 2: Divide using long division method (3x^3+4x^2-7x-5)/(3x-2)

Solution: This problem is also considered "nice" just like the first one, because both the dividend and divisor are in standard forms.

This time around you are dividing a polynomial with four terms by a binomial. Remember that example 1 is a division of polynomial with three terms (trinomial) by a binomial. Hopefully you see the slight difference.

Let's go ahead and work this out!

Description of step Step by step solution by Long Method

1) Focus on the left most terms of both the dividend and divisor.

2) Divide the left most term of the dividend by the left most term of the divisor.

3) Place the partial answer on top.

 

divide 3x^3 by 3x-2, get x^2
   

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

5) Place their product under the dividend.

  • Make sure to align them by similar terms.
distribute x^2 to 3x-2; get 3x^3-2x^2
   
6) Perform subtraction by alternating the signs of the bottom polynomial. switch signs, now we have -3x^3+2x^2
   

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

8) Carry down the next adjacent "unused" term of the dividend

get 6x^2-7x
   

9) Take the left most term of the bottom polynomial, and divide by the left most term of the divisor.

10) Place the answer on top, as usual.

divide (6x^2-7x) by (3x-2), place +2x on top
   

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

distribute 2x to (3x-2); get 6x^2-4x
   

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

13) Carry down the last unused term of the dividend. We're almost there!

 

(6x^2-7x)+(-6x^2+4x)=(-3x-5)
   
14) We are going up one more time. Divide the leading term of the bottom polynomial by the leading term of divisor.  Place the answer up there! (-3x-5)/(3x-2)=-1, place on top
   
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". -1×(3x-2)=3x+2
   
16) Finish this off by subtraction leaving as with a remainder of −7. (-3x-5)-(3x+2)=-7 (remainder)
   

17) Write the final answer in the form...

quotient±(remainder/divisor)

 

x^2+2x-1-[7/(3x-2)]

 

   

 


Example 3: Divide using long division method (3x^4-12x+5)/(x+1)

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 (3x^4+0x^3+0x^2-12x+5)/(x+1). Now all x's are accounted for!

Description of step Step by step solution by Long Method

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

2) Divide the first term of the dividend by the first term of the divisor.

3) Position the partial answer on top.

 

3x^4/(x+1)=3x^3; place partial quotient on top
   

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

5) Put the result under the dividend.

  • Make sure to align them by similar terms.
(3x^3)*(x+1)=3x^4+3x^3
   
6) Subtract them together by making sure to switch the signs of the bottom terms before adding. (3x^4+0x^3)+(-3x^4+3x^3)=-3x^3
   

7) Carry down the next unused term of the dividend.

-3x^3+0x^2
   
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. (-3x^3+0x^2)/(x+1)=-3x^2 (place on top)
   

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

10) Place the answer below then perform subtraction.

(-3x^3+0x^2)+(3x^3+3x^2)=3x^2

   

11) Bring down the next adjacent term of the dividend.

3x^2-12x
   
12) Go up again by dividing the leading term below by the leading term of the divisor. 3x^2/x=3x (place on top)
   

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

  • I believe the pattern makes sense now. Yes?
(3x^2-12x)+(-3x^2-3x)=-15x
   
14) Carry down the last term of the dividend. -15x+5
   
15) Go up again while performing division. -15x/x=-15 (place on top)
   
16) Go down again while performing multiplication. -15*(x+1)=-15x-15
   
17) Do the final subtraction, and we are done! Remainder is equal to 20. (-15x+5)+(15x+15)=20 (remainder)
   

18) The final answer in the form below is...

quotient±(remainder/divisor)

3x^3-3x^2+3x-15+[20/(x+1)]
   

 


Example 4: Divide the given polynomial using long division method (-5x^5-1)/(5x+5)

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

I need to rewrite the problem this way to include all exponents of x: (-5x^5+0x^4+0x^3+0x^2+0x-1)/(5x+5).

     

    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.

-x^4+x^3-x^2+x-1 with remainder 4

 

So the final answer is -x^4+x^3-x^2+x-1+[4/(5x+5)].


Example 5: Divide the given polynomial using long division method (x^4-3x^3+2x^2-x+1)/(x^2-x+1)

Solution:  We have a polynomial with five terms being divided by a trinomial. Both the dividend and divisor are in standard form, and all powers of the variable x are present. This is wonderful because we can now start solving it.

The solution to this problem is presented in animated picture. Observe each step carefully, and see if you can follow it.

animated solution using long division

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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