#### Transcript Algebraic Long Division

```C2: Chapter 1 Algebra and
Functions
Dr J Frost ([email protected])
Terminology
11 ÷ 4 = 2 πππ 3
?
dividend
?
divisor
quotient
?
?
remainder
Normal Long Division
38.
11 4 2 3 . 0 0 0 0
33
93
88
5 0
1. We found how many whole
number of times (i.e. the
quotient) the divisor went into
the dividend.
2. We multiplied the quotient
by the dividend.
3. β¦in order to find the
remainder.
4. Find we βbrought downβ the
next number.
2
6x
x+5
- 2x + 3
6x3 + 28x2 β 7x + 15
3
2
6x + 30x
2
β 2x β 7x
β 2x2 β 10x
3x + 15
3x + 15
0
The Anti-Idiot Test:
You can check your solution by
finding (x+5)(6x2 β 2x + 3)
2
3x
x-1
+ 0 β4
3x3 β 3x2 β 4x + 4
3
2
3x β 3x
0 β 4x + 4
β 4x + 4
0
2
2x
+ 3x β 4
x - 4 2x3 β 5x2 β 16x + 10
3
2
2x β 8x
Find the
2
3x β 16x
remainder.
3x2 β 12x
-4x + 10
Q: Is (x-4) a factor of
-4x
+
16
2x3 β 5x2 β 16x + 10?
-6
Exercises
1a
1i
2a
2i
2b
Exercise 1B
Divide π₯ 3 + 6π₯ 2 + 8π₯ + 3 by π₯ + 1
? +3
π₯ 2 + 5π₯
Divide π₯ 3 β 8π₯ 2 + 13π₯ + 10 by π₯ β 5
? β2
π₯ 2 β 3π₯
Divide 6π₯ 3 + 27π₯ 2 + 14π₯ + 8 by π₯ + 4
6π₯ 2 + 3π₯
? +2
Divide β5π₯ 3 β 27π₯ 2 + 23π₯ + 30 by π₯ + 6
β5π₯ 2 +? 3π₯ + 5
Exercise 1C
Find the remainder when π₯ 3 + 4π₯ 2 β 3π₯ + 2 is
divided by π₯ + 5.
β8
?
Dividing polynomials with βmissingβ terms
Divide x3 β 1 by x β 1
How would we write the division?
2
π΄ππ π€ππ: π₯ + ?π₯ + 1
In general, the difference of two cubes can be factorised as:
π₯ 3 β π¦ 3 = π₯ β π¦ π₯ 2 + π₯π¦ + π¦ 2
Dividing polynomials with βmissingβ terms
Divide x4 β 16 by (x+2)
π΄ππ π€ππ: π₯ 3 β 2π₯ 2 ?+ 4π₯ β 8
Recap
dividend
quotient
8
= 2+
3
divisor
1
3
remainder
Remainder and Factor Theorem
Weβre trying to work out the remainder when
we divide a polynomial π π₯ by π₯ β π
π π₯
π
=π π₯ +
π₯βπ
π₯βπ
π(π₯) = (π₯ β π)π(π₯) + π
So what does f(a) equal?
What if π π = 0?
Remainder and Factor Theorem
!
Remainder Theorem
For a polynomial π(π₯), the remainder when
π(π₯) is divided by π₯ β π is π π .
!
Factor Theorem
If π π = 0, then by above, the remainder is
0. Thus (π₯ β π) is a factor of π π₯ .
Basic Examples
Remainder when π₯ 2 + 1 is divided by π₯ β 2?
π 2 = ?5
Remainder when π₯ 3 β π₯ is divided by π₯ + 1?
π β1 =?0
Remainder when π₯ 2 + 1 is divided by 2π₯ β 1?
1
5
π
=?
2
4
Remainder when π₯ 2 β π₯ is divided by 3π₯ + 4?
4
28
π β =?
3
9
Examples
Show that (x β 2) is a factor of x3 + x2 β 4x - 4
π 2 = 8 + 4? β 8 β 4 = 0
Examples
Fully factorise 2x3 + x2 β 18x β 9
= (π₯ β 3)(π₯ +?3)(2π₯ + 1)
Tip: If f(x) = 2x3 + x2 β 18x β 9, then try f(-1), f(1), f(2), etc. until
one of these is equal to 0.
Examples
Fully factorise π₯ 3 + 6π₯ 2 + 5π₯ β 12
= (π₯ β 1)(π₯ +? 3)(π₯ + 4)
Given that (π₯ + 1) is a factor of 4π₯4 β 3π₯2 + π,
find the value of π.
π = β1?
Examples
C2 May 2013 (Retracted)
π = π,?
π = βπ
(π β π)(ππ β
? π)(ππ + π)
Examples
Exercise 1D
Q1, 2, 4, 6, 8, 10
Recap
Q10) Given that (π₯ β 1) and (π₯ + 1) are factors of
ππ₯3 + ππ₯2 β 3π₯ β 7 find the value of π and π.
π = 3,?π = 7
Recap
Find the remainder when
16x5 β 20x4 + 8 is divided by (2π₯ β 1)
15
πππππππππ ππ  ?
2
Bro tip: think what you could make x in
order to make the factor (2x-1) zero.
Recap
When 8π₯4 β 4π₯3 + ππ₯2 β 1 is divided by (2π₯ +
1) the remainder is 3. Find the value of π.
π =? 16
Exercises
Le Exercise 1E:
β’ Q1f, g, h, i
β’ 2, 4, 6, 8, 10
Le Exercise 1F
β’ 4, 5, 8, 10, 15.
```