Synthetic Division
Synthetic Division - How it Works - Video
Synthetic Requirements
Synthetic Requirements:
We have two requirements. The first one is that the divisor must be degree 1 (linear factor), x +3 or x - 5.
The second one is the coefficient of the divisor must be 1. So 3x + 5 is degree 1, but the coefficient is 3 while the constant is 5. We can change it so that we can use synthetic division. So we have to rearrange some the numbers. Take a look at example 3 for my detail.
Example 1
Example 1:
Let's first use long division to find our answer so we can see what is happening when we use synthetic. It is important to understand why you are doing something instead of just repeating the process.
We have our dividend, f(x) = x3 - 2x2 - 5x + 6 and our divisor, x - 4.
After completing the long division we have our quotient, x2 + 2x + 3 and our remainder, 18.
The first step is write down our c. We have x - 4 so c = 4 because (x - 4) ⇔ (x - c).
Our next step is write down the coefficients of each term. We don't have to write any empty zeros since we have each term.
Above the coefficients, we have written the term that the column represents. Notice that the power is one less than our biggest power in our function, x3. Then we subtract one until we have x0 (remember anything to the 0 power is 1, so we won't have an x in our answer for this term) and the last column is the remainder.
So we bring down the first coefficient 1. Then we multiply 1 and 4 => 4, and write down the result under the second coefficient.
Now we add -2 and 4 => 2. Then we multiply 2 and 4 => 8.
Now we add -5 and 8 => 3. Then we multiply 3 and 4 => 12.
Now we add 6 and 12 => 18. We stop here since we don't have more columns.
We have our quotient the first three terms and our remainder the last term. So our quotient is x2 + 2x + 3 and our remainder is 18.
So f(x) = x3 - 2x2 - 5x + 6 divided by x - 4 is x2 + 2x + 3 + 18/ (x - 4).
Here we have our model, where we drop, multiply then add, and repeat.
Example 2
Example 2:
Let's first use long division to find our answer so we can see what is happening when we use synthetic. It is important to understand why you are doing something instead of just repeating the process.
We have our dividend, f(x) = x3 - 8x - 5 and our divisor, x + 2.
After completing the long division we have our quotient, x2 - 2x - 4 and our remainder, 3.
The first step is write down our c. We have x + 2 so c = -2 because (x + 2) ⇔ (x - [-2]) ⇔ (x - c).
Our next step is write down the coefficients of each term. In this case, we are missing a term. We don't a term with a degree 2 so we need to put a place holder 0 for the second column after the 1.
Above the coefficients, we have written the term that the column represents. Notice that the power is one less than our biggest power in our function, x3. Then we subtract one until we have x0 (remember anything to the 0 power is 1, so we won't have an x in our answer for this term) and the last column is the remainder.
So we bring down the first coefficient 1. Then we multiply 1 and -2 => -2, and write down the result under the second coefficient.
Now we add 0 and -2 => -2. Then we multiply -2 and -2 => 4.
Now we add -8 and 4 => -4. Then we multiply -4 and -2 => 8.
Now we add -5 and 8 => 3. We stop here since we don't have more columns.
We have our quotient the first three terms and our remainder the last term. So our quotient is x2 - 2x - 4 and our remainder is 3.
So f(x) = x3 - 8x - 5 divided by x + 2 is x2 - 2x - 4 + 3 / (x + 4).
Example 3
Example 3:
Let's first use long division to find our answer so we can see what is happening when we use synthetic. It is important to understand why you are doing something instead of just repeating the process.
We have our dividend, f(x) = x3 - 3x2 - 10x + 24 and our divisor, 2x - 3.
After completing the long division we have our quotient, (1/2) * x2 - (3/4) * x - (49/8) and our remainder, 45/8.
The first step is write down our c. We have 2x - 3 so c = 3/2 because (2x - 3) ⇔ (x - [3/2]) ⇔ (x - c).
Our next step is write down the coefficients of each term. In this case, we are missing a term. We don't have to write any empty zeros since we have each term.
Above the coefficients, we have written the term that the column represents. Notice that the power is one less than our biggest power in our function, x3. Then we subtract one until we have x0 (remember anything to the 0 power is 1, so we won't have an x in our answer for this term) and the last column is the remainder.
So we bring down the first coefficient 1. Then we multiply 1 and 3/2 => 3/2, and write down the result under the second coefficient.
Now we add -3 and 3/2 => -3/2. Then we multiply -3/2 and 3/2 => -9/4.
Now we add -10 and -9/4 => -49/4. Then we multiply --49/4 and 3/2 => -147/8.
Now we add 24 and -147/8 => 45/8. We stop here since we don't have more columns.
We have our quotient the first three terms and our remainder the last term. So our quotient is x2 - (3/2) * x - (49/4) and our remainder is 45/8.
So f(x) = x3 - 3x2 - 10x + 24 divided by 2x - 3 is x2 - (3/2) * x - (49/4) + (45/8) / (2x - 3).
Wait a second. We don't have the same as we worked in with the long division.
Let's take a closer look at the why it didn't work the first time.
When we multiplied our factor by 1/2 so our coefficient would be 1 so we could use synthetic division. We forgot to multiply the dividend by 1/2. What we do to the top we need to do the bottom.
Here we multiply the coefficients by 1/2.
Now, we have our quotient the first three terms and our remainder the last term. So our quotient is (1/2) * x2 - (3/4) * x - (49/8) and our remainder is 45/16.
So f(x) = x3 - 3x2 - 10x + 24 divided by 2x - 3 is (1/2) * x2 - (3/4) * x - (49/8) + 45 / [16 * (2x - 3)].
Wait a second. Our coefficients are the same, but our remainder is different. Well our remainder is being divided by (x - [3/2]) not (2x -3).
Here we need to simplify our factor by multiplying by 2 to the top and 2 to the bottom. Now our remainder looks like our answer that we find with long division.
One way to have fewer fractions is to leave the coefficients the same and to multiply at the end. Just be careful to only the first terms and leave the last one, which is the remainder, because we didn't multiply to find it.
So for this example, we would multiply 1, -3/2, -49/4 by 1/2 and leave 49/8, the remainder, alone. After doing that we would get the same answer.