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Related Lessons: Adding and Subtracting Polynomials Multiplying Polynomials Polynomial Long Division

Synthetic Division | Step by Step

 

I must say that synthetic division is the most "fun" way of dividing polynomials.  It has lesser steps to arrive at the answer as compared to polynomial long division method. In this lesson, I will go over five (5) examples that should hopefully make you familiar with the basic procedures in successfully dividing polynomials using synthetic division.

After going over the worked examples below, you may do some extra practice using the pre formatted synthetic division worksheets with answers. Enjoy!

 

Example 1: (x^4-3x^3-11x^2+5x+17)/(x+2) See solution (animated picture)
Example 2: (x^5-3x^3-4x-1)/(x-1) See solution (animated picture)
Example 3: (-2x^4+1)/(x-3) See solution (animated picture)
Example 4: (-x^5+1)/(x+1) See solution (animated picture)
Example 5: (2x^3-13x^2+17x-10)/(x-5) See solution (animated picture)

 

Things to remember before proceeding with the actual steps of synthetic division.

  • Make sure the dividend is in standard form. That means the powers are in decreasing order.
  • The divisor must be in the form x-(c)

 


Example 1: Divide using Synthetic Division Method (x^4-3x^3-11x^2+5x+17)/(x+2)

Let us re-examine the given problem and make the necessary adjustments, if necessary.

The dividend (stuff to divide) is in standard form because the exponents are in decreasing order. That's good!

The divisor needs to be rewritten as x-(c) or x-(-2) therefore c=-2

 

(x^4-3x^3-11x^2+5x+17) is the dividend while (x+2) is the divisor

 

At this point, I can now setup the synthetic division by extracting the coefficients of the dividend and then lining them up on top.

Directly to the left side, place the value of c = −2 inside the "box".

Finally, construct a horizontal line just below the coefficients of the dividend.

 

  • (If the animated picture doesn't work, please refresh click refresh icon your browser)

 

See animated solution below

animated gif solution to synthetic division example 1: 1 -5 -1 7 3

 

Steps:

1. Drop the first coefficient below the horizontal line.

2. Multiply that number you drop by the number in the "box". Whatever its product, place it above the horizontal line just below the second coefficient.

3. Add the column of numbers, then put the sum directly below the horizontal line.

4. Repeat the process until you run out of columns to add.

 

The last number below the horizontal line is always the remainder! The remainder in this problem is 3.

So how do we present our final answer? 

remainder is 3  

Show your final answer in the form

x^3-5x^2-x+7+[3/(x+2)

Notice that the numbers below the horizontal line except the last (remainder) are the coefficients of the Quotient. 

More so, the exponents of the variables of the quotient are all reduced by 1.

 


Example 2: Divide using Synthetic Division Method (x^5-3x^3-4x-1)/(x-1)

This is not a trick question. Maybe you will see some problems like this in your study about synthetic division. Notice that the quotient does not have all the exponents of variable x.

I can see that we are missing x^4 and x^2. To include all the coefficients of variable x in decreasing power, we should rewrite the original problem like this. Attach zeroes on those missing x's. Also express the divisor as x-(c)which clearly reveals the value of c, that is, c = +1.

(x^5+0x^4-3x^3+0x^2-4x-1)/[x-(+1)]

From this point, I can now setup the numbers ready for synthetic division process.

 

  • (If the animated picture doesn't work, please refresh click refresh button your browser)

 

See animated solution below

animated gif synthetic division example 2: 1 1 -2 -2 -6 -7

 

Steps:

1. Drop the first coefficient below the horizontal line.

2. Multiply that number you drop by the number in the "box". Whatever its product, place it above the horizontal line just below the second coefficient.

3. Add the column of numbers, then put the sum directly below the horizontal line.

4. Repeat the process until you run out of columns to add.

So putting the final answer in the form quotient±(remainder/divisor), we have x^4+x^3-2x^2-2x-6-[7/(x-1).


Example 3: Divide using Synthetic Division Method (-2x^4+x)/(x-3)

This is becoming more interesting! The quotient definitely looks horrible because it is missing a lot. Not only it lacks some x's which are x^3and x^2 but the constant is also gone.

To fix this, I will rewrite the original problem in such a way that all x's are accounted for. But more importantly, do not forget to include the missing constant which is zero.

The "new and improved" problem should look like this: (-2x^4+0x^3+0x^2+x+0)/[x-(+3)]. From here, proceed with the steps of synthetic division as usual.

 

  • (If the animated picture doesn't work, please refresh refresh icon your browser)

 

See animated solution below

animated gif solution to synthetic division example 3: -2 -6 -18 -53 -159

 

Steps:

1. Drop the first coefficient below the horizontal line.

2. Multiply that number you drop by the number in the "box". Whatever its product, place it above the horizontal line just below the second coefficient.

3. Add the column of numbers, then put the sum directly below the horizontal line.

4. Repeat the process until you run out of columns to add.

Okay then, the final answer for this is -2x^3-6x^2-18x-53-[159/(x-3)].

You can write the final answer in two ways. The first one is using the minus or subtraction symbol to indicate that the remainder is negative. The second one is using the + symbol but attaching a negative symbol on the numerator. They mean the same thing!

 


Example 4: Divide using Synthetic Division Method (-x^5+1)/(x+1)

Don't be discouraged by this problem. This is actually quite easy especially now that you have gone through a few examples already. Always remember to "fill in the missing parts", right?

Observe the dividend and you should agree that the missing parts are x^4, x^3, x^2and x.

Rewriting the original problem that is synthetic-division ready, we get...

(-x^5+0x^4+0x^3+0x^2+0x+1)/[x-(-1)]

 

We populated the missing x's with zeroes and explicitly solve for c= −1.

Hopefully you find it enjoyable going through each step of synthetic division. Here we go again!

 

  • (If the animated picture doesn't work, please refresh refresh button your browser)

 

See animated solution below

animinated synthetic division example problem 4: -1 1 -1 1 -1 2

 

Steps:

1. Drop the first coefficient below the horizontal line.

2. Multiply that number you drop by the number in the "box". Whatever its product, place it above the horizontal line just below the second coefficient.

3. Add the column of numbers, then put the sum directly below the horizontal line.

4. Repeat the process until you run out of columns to add.

The last number below the horizontal line will always be the remainder. Don't forget that. In this case, the remainder equals 2.

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

 


Example 5: Divide using Synthetic Division Method (2x^3-12x^2+17x-10)/(x-5)

In this example, we get a remainder of zero after applying the steps in synthetic division. When that happens the divisor becomes a factor of the dividend. In other words, the divisor evenly divides the dividend.

By examining the problem, I see that there are no missing components. All powers of x's are accounted for, and we have a constant. That's great! This problem is in fact synthetic-division ready.

 

  • (If the animated picture doesn't work, please refresh click refresh your browser)

 

See animated solution below

animated gif solution for synthetic division example 5: 2 -3 2 0

 

Steps:

1. Drop the first coefficient below the horizontal line.

2. Multiply that number you drop by the number in the "box". Whatever its product, place it above the horizontal line just below the second coefficient.

3. Add the column of numbers, then put the sum directly below the horizontal line.

4. Repeat the process until you run out of columns to add.

 

Because the remainder equals zero, this means the divisor x-5 is a factor of the dividend 2x^3-13x^2+17x-10

therefore (2x^3-13x^2+17x-10)=(x-5)(2x^2-3x+2)

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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