C2: Chapter 1 Algebra and Functions Dr J Frost ([email protected]) Last modified: 1 st September 2015.

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