1. Additional Mathematics for the OCR syllabus - Algebra 8
AS Mathematics
Algebra – Long division
Lesson A8
This lesson focuses on long division. Students will need to have excellent algebraic manipulation skills and be very confident about working with negative numbers if they are to make progress with this lesson.
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

2. Additional Mathematics for the OCR syllabus - Algebra 8
Objectives
Be able to divide polynomials using algebraic long division
Be able to factorise polynomials up to order 3 given one linear factor
An understanding of numerical long division is assumed, but we do recap this quickly!
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

3. Additional Mathematics for the OCR syllabus - Algebra 8
Long Division
Let’s start by looking at a numerical example!
We need to set it out the ‘old fashioned’ way
Example 1
Divide 364 by 13
13 goes into 36 twice 2 8
remainder 10 (36 – 26 = 10) 13
364
…bring down the 4 - 26
The example is revealed line by line. It’s an idea to ask students to work through the question prior to revealing the solution.
At the very least students should try to predict the next line in the working out before it is revealed. 10 4
13 goes into times -
104
remainder 0 (104 – 104 = 0)
ANSWER: 28
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

4. Additional Mathematics for the OCR syllabus - Algebra 8
Example 2
16 goes into 57 3 times
572 divided by 16
3 x 16 = 48 3 5
remainder 9 (57 – 48 = 10) 16
572
…bring down the 2 - 48
16 goes into 92 5 times 9 2
5 x 16 = 80 - 80
Again this question should be revealed line by line, with the students predicting the next step in the solution.
Point out that 35 x = 572.
remainder 12 (92 – 80 = 12) 12
ANSWER: 3512/16
or 35 ¾
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

5. Additional Mathematics for the OCR syllabus - Algebra 8
Let’s try with algebra!
Example 3
Look at the second term in both expressions
Look at the first term in both expressions
What is x2 ÷ x ?
Find
(x2 + x - 2) ÷ (x + 1)
x goes into x2, x times
x(x – 1)
= x2 - x
+ 2 x
remainder 2x
x - 1
x2 + x - 2
…bring down the - 2 -
x2 - x
Go through the first example using algebra step by step, the circles should help to point out the important parts of the question.
- 2 2x
x goes into 2x twice -
2x - 2
2(x – 1)
= 2x - 2
remainder 0
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

6. Additional Mathematics for the OCR syllabus - Algebra 8
So (x2 + x - 2) ÷ (x - 1) = x + 2
Notice
quadratic ÷ linear => linear
What if we multiply (x - 1) by (x + 2) ?
(x - 1)(x + 2)
= x2 + x - 2
Check your answer by reversing the question & multiplying out the brackets!
We found that (x – 1) goes into x2 + x – 2 exactly (x + 2) times ...
... (x – 1) and (x + 2) are the factors of x2 + x – 2
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

7. Additional Mathematics for the OCR syllabus - Algebra 8
Example 4
Look at the first term in both expressions
Look at the second term in both expressions
What is x3 ÷ x ?
Divide the cubic expression x3 + 2x2 - 5x - 6
by x + 3
x goes into x3, x2 times
x2(x + 3)
= x3 + 3x2 x2
- x
- 2
remainder - x2
x + 3
x3 + 2x2 - 5x - 6
…bring down the - 5x -
(x3 + 3x2)
x goes into -x2, - x times
- x2
- 5x
- x(x + 3)
= - x2 - 3x -
- x2 - 3x
The example is revealed line by line.
Get the pupils to predict the next line of the solution & discuss alternative methods.
remainder - 2x
- 2x
- 6
…bring down the - 6 -
(- 2x - 6)
x goes into -2x, - 2 times
- 2(x + 3)
= - 2x - 6
remainder 0
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

8. Additional Mathematics for the OCR syllabus - Algebra 8
So (x3 + 2x2 - 5x - 6) ÷ (x + 3) = x2 - x - 2
Additional Mathematics for the OCR syllabus - Algebra 8
Notice
cubic ÷ linear => quadratic
x3 + 2x2 - 5x - 6
(x + 3)(x2 - x - 2)
(x + 3) and (x2 – x – 2) are factors of x3 + 2x2 - 5x – 2
BUT (x2 - x - 2) can be factorised.
(x2 - x - 2) = (x + 1)(x - 2)
Stress the importance of checking the answer by expanding the brackets.
So x3 + 2x2 - 5x - 6 = (x + 3)(x + 1)(x - 2)
x3 + 2x2 - 5x – 2 has 3 linear factors
(x + 3), (x + 1) and (x - 2)
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

9. Additional Mathematics for the OCR syllabus - Algebra 8
Questions
1. Divide:
a) 2x3 - 3x2 - 3x by x - 2
2x3 + 3x by 2x - 1
2x3 - 11x2 + 12x by x - 5
2. Factorise the above expressions completely
Extra practice on this topic would be beneficial.
Either use these questions or look at the Additional mathematics for OCR textbooks, or any A’ level textbook for questions.
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

10. Additional Mathematics for the OCR syllabus - Algebra 8
Answers 1.
a) 2x2 + x - 1
x2 + 2x + 1
2x2 - x + 7 2.
(x - 2)(2x2 + x - 1) = (x - 2)(2x - 1)(x + 1)
(2x - 1)(x2 + 2x + 1) = (2x - 1)(x + 1)2
(x - 5)(2x2 – x + 7)
Give instant feedback on these questions by revealing the answers.
Written by HVaughan (North Chadderton) and LDobson (Blue Coat)