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?

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: Divide using long division method

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. 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. 6) Perform subtraction by switching the signs of the bottom polynomial. 7) Proceed with regular addition vertically. Notice that the first column from the left cancels each other out. Nice! 8) Carry down the next adjacent "unused" term of the dividend. 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. 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. 13)  Subtraction means you will switch the signs (in red). 14) Perform regular addition along the columns of similar terms 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.

Example 2: Divide using long division method

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!

Example 3: Divide using long division method

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!

Example 4: Divide the given polynomial using long division method

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: .

 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.

So the final answer is .

Example 5: Divide the given polynomial using long division method

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.

 Practice Problems with Answers Worksheet 1 Worksheet 2

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