C2: Chapter 1 Algebra and Functions
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 1st September 2015
Terminology
11÷4=2𝑟𝑒𝑚 3
dividenddivisor
quotient
remainder
??
?
?
4 2 3 . 0 0 0 0113
3 3 9 3
8 .
8 85 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. …in order to find the remainder.
4. Find we ‘brought down’ the next number.
Normal Long Division
x + 5 6x3 + 28x2 – 7x + 156x2
6x3 + 30x2
– 2x2 – 7x
- 2x
– 2x2 – 10x3x + 15
+ 3
3x + 15 0
The Anti-Idiot Test:You can check your solution by finding (x+5)(6x2 – 2x + 3)
x - 1 3x3 – 3x2 – 4x + 43x2
3x3 – 3x2
0 – 4x + 4
+ 0
– 4x + 4 0
– 4
x - 4 2x3 – 5x2 – 16x + 102x2
2x3 – 8x2
3x2 – 16x
+ 3x
3x2 – 12x-4x + 10
– 4
-4x + 16 -6
Find the remainder.
Q: Is (x-4) a factor of 2x3 – 5x2 – 16x + 10?
Exercises
Exercise 1BDivide by
Divide by
Divide by
Divide by
Exercise 1CFind the remainder when is divided by .
1a
1i
2a
2i
2b
?
?
?
?
?
Divide x3 – 1 by x – 1
How would we write the division?
Dividing polynomials with ‘missing’ terms
𝐴𝑛𝑠𝑤𝑒𝑟 :𝑥2+𝑥+1?
For Olympiad enthusiasts:In general, the difference of two cubes can be factorised as:
Divide x4 – 16 by (x+2)
Dividing polynomials with ‘missing’ terms
𝐴𝑛𝑠𝑤𝑒𝑟 :𝑥3−2𝑥2+4 𝑥−8?
8
3= 2 +
13
dividend
divisor
quotient remainder
Recap
We’re trying to work out the remainder when we divide a polynomial by
𝑓 (𝑥)=(𝑥−𝑎)𝑞 (𝑥 )+𝑟So what does f(a) equal?
What if ?
Remainder and Factor Theorem
𝑓 (𝑥 )𝑥−𝑎
=𝑞 (𝑥 )+ 𝑟𝑥−𝑎
Remainder and Factor Theorem
Remainder TheoremFor a polynomial , the remainder when is divided by is .
Factor TheoremIf , then by above, the remainder is 0. Thus is a factor of .
!
!
Basic Examples
Remainder when is divided by ?
Remainder when is divided by ?
Remainder when is divided by ?
Remainder when is divided by ?
?
?
?
?
Show that (x – 2) is a factor of x3 + x2 – 4x - 4
Examples
𝑓 (2 )=8+4−8−4=0?
Fully factorise 2x3 + x2 – 18x – 9
Tip: If f(x) = 2x3 + x2 – 18x – 9, then try f(-1), f(1), f(2), etc. until one of these is equal to 0.
Examples
¿ (𝑥−3)(𝑥+3)(2𝑥+1)?
Examples
Fully factorise
?
Given that is a factor of , find the value of .
𝑎=−1?
ExamplesC2 May 2013 (R)
𝒂=𝟔 ,𝒃=−𝟒
(𝒙−𝟐)(𝟐 𝒙−𝟏)(𝟑 𝒙+𝟐)
?
?
Exercise 1DQ1, 2, 4, 6, 8, 10
Examples
Q10) Given that and are factors of find the value of and .
Recap
𝑝=3 ,𝑞=7?
Find the remainder when 16x5 – 20x4 + 8 is divided by
Bro tip: think what you could make x in order to make the factor (2x-1) zero.
Recap
𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 𝑖𝑠152
?
When is divided by the remainder is 3. Find the value of .
Recap
𝑎=16?
Le Exercise 1E: • Q1f, g, h, i• 2, 4, 6, 8, 10
Le Exercise 1F• 4, 5, 8, 10, 15.
Exercises