Algebraic Division Summary
Academic Skills Advice
It’s a good idea to practice basic long division before moving on to algebraic division. Try
some questions and make sure you remember the method.
Algebraic Division:
Divide 𝒙𝟑 + 𝟔𝒙𝟐 − 𝟕𝒙 − 𝟔𝟎 by 𝒙 + 𝟒
Set up the division:
(𝒙 + 4) | 𝒙𝟑 + 6𝑥 2 − 7𝑥 − 60
Divide the 1st term by the 𝑥 and put the answer on the answer line.
𝒙𝟐
𝑥3 ÷ 𝑥 = 𝑥2:
(𝒙 + 4) | 𝒙𝟑 + 6𝑥 2 − 7𝑥 − 60
Next multiply the answer (𝑥 2 ) by the divisor (𝑥 + 4) and subtract from the original:
𝒙𝟐
(𝒙 + 𝟒) | 𝑥 3 + 6𝑥 2 − 7𝑥 − 60
2
𝑥 × (𝑥 + 4):
𝒙𝟑 + 𝟒𝒙𝟐
Subtract:
0 + 2𝑥 2
Bring the next term (−7𝑥) down to join the (2𝑥 2 ) and repeat the whole process until you get
to the end:
𝑥2
(𝒙 + 4) | 𝑥 3 + 6𝑥 2 − 7𝑥 − 60
𝑥 3 + 4𝑥 2
Bring the −7𝑥 down:
𝟐𝒙𝟐 − 7𝑥
Repeat the whole process: 2𝑥 2 ÷ 𝑥 = 2𝑥, put this on the answer line then do 2𝑥 × (𝑥 + 4)
and subtract............... continue the process:
𝑥 2 + 2𝑥 − 15
(𝑥 + 4) | 𝑥 3 + 6𝑥 2 − 7𝑥 − 60
𝑥 3 + 4𝑥 2
2𝑥 2 − 7𝑥
2𝑥 2 + 8𝑥
−15𝑥 − 60
−15𝑥 − 60
0
There is no
remainder
We have worked out that (𝒙𝟑 + 𝟔𝒙𝟐 − 𝟕𝒙 − 𝟔𝟎) ÷ (𝒙 + 𝟒) = 𝒙𝟐 + 𝟐𝒙 − 𝟏𝟓
n.b. if there is a missing term in the expression you are dividing into then you need to
include the term with a zero:
e.g. 𝒙𝟑 + 𝟓𝒙 + 𝟐 should be written as 𝒙𝟑 + 𝟎𝒙𝟐 + 𝟓𝒙 + 𝟐
© H Jackson 2013 / 2015 / Academic Skills
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Ruffini’s Rule:
Ruffini’s Rule is another method of doing algebraic division:
It is used for dividing a polynomial by (𝑥 − 𝑟)
E.g. divide 𝟐𝒙𝟑 + 𝟑𝒙𝟐 − 𝟒 by (𝒙 + 𝟏)
We have to divide by (𝑥 − 𝑟) so we use (𝑥 − (−1))
Step 1: Set up a table as follows:
Coefficients
𝑟
𝑥3
𝑥2
𝑥
𝑁𝑢𝑚𝑏𝑒𝑟
2
3
0
−4
−1
Step 2: Bring the 1st coefficient down:
𝑥3
𝑥2
𝑥
𝑁𝑢𝑚𝑏𝑒𝑟
2
3
0
−4
−1
𝟐
Step 3: do 𝒓 × 1st coefficient:
𝑥3
𝑥2
𝑥
𝑁𝑢𝑚𝑏𝑒𝑟
2
3
−𝟐
0
−4
𝑥3
𝑥2
𝑥
𝑁𝑢𝑚𝑏𝑒𝑟
2
3
−2
𝟏
0
−4
−1
𝑟 × 1st coefficient (−1 × 2)
2
Step 4: 𝟑 − 𝟐 = 𝟏
−1
2
© H Jackson 2013 / 2015 / Academic Skills
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Step 5: repeat the process (from step 3) until you complete the table:
𝑥3
𝑥2
𝑥
𝑁𝑢𝑚𝑏𝑒𝑟
2
3
−2
1
0
−1
−1
−4
1
−3
−1
2
The answer: the final line gives the coefficients of the answer (remember the answer
will be 1 order less than the question):
𝑥3
𝑥2
2
𝑥
𝑁𝑢𝑚𝑏𝑒𝑟
3
0
−2 −1
𝟏
−𝟏
−1
𝟐
−4
1
−𝟑
Coefficients of answer
2𝑥 2 + 𝑥 − 1 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 − 3
Answer:
Therefore 𝟐𝒙𝟑 + 𝟑𝒙𝟐 − 𝟒 ÷ (𝒙 + 𝟏) = 𝟐𝒙𝟐 + 𝒙 − 𝟏 𝒓𝒆𝒎𝒂𝒊𝒏𝒅𝒆𝒓 − 𝟑
Note that Ruffini’s Rule can be used even when the coefficient of r is greater than 1. Just
divide everything by the coefficient of r then proceed as normal.
E.g. divide 𝟑𝒙𝟑 − 𝟖𝒙𝟐 + 𝟐𝟓𝒙 − 𝟏𝟒 by (𝟑𝒙 − 𝟐)
So that we can see what’s happening let’s write the calculation as a fraction:
𝟑𝒙𝟑 −𝟖𝒙𝟐 +𝟐𝟓𝒙−𝟏𝟒
𝟑𝒙−𝟐
Now factorise out the 3 so that the coefficient of r=1
𝟖
𝟑
𝟐𝟓
𝟑
𝟐
𝟑(𝒙− )
𝟑
𝟏𝟒
𝟑
𝟑(𝒙𝟑 − 𝒙𝟐 + 𝒙− )
The 3 will cancel at the top and bottom leaving us with the calculation:
𝟖
(𝒙𝟑 − 𝟑 𝒙𝟐 +
𝟐𝟓
𝟑
𝒙−
𝟏𝟒
𝟑
𝟐
) ÷ (𝒙 − 𝟑)
Now we can use Ruffini’s Rule as above.
Try it………………. (you should get the answer 𝑥 2 − 2𝑥 + 7)
© H Jackson 2013 / 2015 / Academic Skills
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