Products and rules

1. Multiply a binomial by a trinomial2. Highest common factor3. Difference of two squares4. Factorise trinomials5. Factorise by grouping6. Simplify algebraic fractions with monomial

denominators

Multiply a binomial by a trinomial

Theory of Products: These rules form the basis of

factorisation, which is the reverse of multiplication.

Multiply a binomial by a trinomial:

Always have the binomial as the first factor and then multiply all three terms in the second factor with the first term of the binomial. Then the second term with all three terms of the trinomial and then simplify.

2 2 2

2 2

a( b+c)=ab+bc........distributive law

( a+b) ( c+d) = ac + ad+bc+bd

( a+b+c) ( p+q)=ap+aq+bp+dq

( a±b) =a ±2ab+b

( a+b) ( a-b)=a -b

a

+c

a + b2a ab

ac bc

2 2

3 2 2 2 2 3

3 2 2

3 6 3 2 4 2

a b a ab b

a a b ab a b ab b

Examples

2

2 2

3 2 2

3 2

1. 3 3 2 4

3 2 4 3 3 2 4

3 2 4 9 6 12

3 7 10 12

x x x

x x x x x

x x x x x

x x x

2

3 2 2

3

2. 2 1 4 2 1

8 4 2 4 2 1

8 1

x x x

x x x x x

x

Test your knowledgeQuestion 1

Simplify ( 3x +1)(9x2 – 3x +1)

Answer

A) 27x3 + 2x2 - 1 B) 9x3 – 2x2 +3x +1C) 27x3 +1 D) None

Test your knowledge

Question 2

Simplify : (5x-1) (x2 +2x – 8)

Answer

A) 5x3 +9x2 - 35x + 8 B) 6x3 +32x – 8C) 5x3 +5x2 - 35x + 8 D) 5x3 +9x2 - 42x + 8

Factorisation

Steps for factorisation

Look fora) Highest Common Factor (HCF)b) Difference of two Squaresc) Trinomiald) Sum and Difference of two cubese) If there is brackets multiply out and rearrange from (1 –

4)

Highest common factor

Examples

Factorise

1) p2q2 – pq2

=pq2 (p – 1)

2) 3a(p – 3q) – 2b(p – 3q)=(p – 3q)(3a – 2b)

3) 2p(a – b) – (a – b) =(a – b)(2p -1)

Difference of two squares

The difference of two squares and perfect squares1) a2-b2=(a-b)(a+b)

To identify the difference of two squares the following must be possible:

1. √T1

2. √ T2

3. There must be a minus sign between the two terms.

Examples1)

2)

3)

4)

Factorise trinomials

2 5 6m m An expression in which the highest exponent is 2 is a quadratic expression. is called a trinomial because it contains three terms. If we factorise this quadratic trinomial, we get two binomials as factors:

23 2 5 6m m m m

multiplication

Factorisation

Factorisation is the reverse of multiplication

Factorise trinomials

The sign of the last term indicates that the signs inside the two brackets will be the same e.g. the “+”sign in:

Write down the brackets with the correct signs. In this example both have to be negative.

Now check the factors of the constant term of which the sum add to 6, then choose the correct combination and factorise

2 5 6m m

m m

6 1 6 sum of factors: 1+ 6 = 7

2 3 sum of factors: 2 + 3 = 6

2 + 3 = 6 are the correct combination

Examples Factorise:

One factor will be negative, one will be positive.

Combinations of factors:

The correct one must give – 8x

Therefore:

x x

3 3, but sum must be = - 9

- 9 1 and sum is - 8

2 8 9x x

1 9x x

2 8 9 1 9x x x x

and this sum = 0

Test your knowledgeQuestion 3

Factorise the following: x2 + 14x +24

Answer

A) (x -2) (x- 12) B) (x + 2) (x + 12)C) (x + 1) (x + 24) D) None of the above

Test your knowledge

Question 4

Factorise the following: x2 – 2x – 24

Answer

A) (x – 4) (x + 6) B) (x – 12) (x + 2)C) (x + 4) (x + 6) D) (x + 4) (x - 6)

Solution

B

Factorise by grouping

If we have more than three terms to factorise, we have to group either two-two or three – one together.

Always first check for a common factor in all four terms.

Sometimes we have to change the order by rearranging when grouping. Always change signs of each term when you place a bracket after a negative sign, because it is like taking out “– 1” as a common factor.

Remember 2 + c = c + 2

4 3 2 3 2

2

2

1

1 1

1 1

x x x x x x x x

x x x x

x x x

3 3 3( ) ( )x p qp qx x p q p x

2 2 4

2 4 2

2 ( 2 ) (2 )

(2 )(2 1)

am n an m

am an n m

a m n n m

n m a

Test your knowledge

Question 5

Factorise the following: 3xy – 3x + 3 – 3y

Answer A) 3x(y – 1) B) 3(y + 1)(x + 1)C) 3x(x + 1)(y – 1) D) 3(y – 1)(x – 1)

Test your knowledge

Question 6

Factorise the following: 4a2 – 2a + 9b2 – 3b + 12ab

Answer

A) (3a + 2b)(3a +2b +1) B) (2a +3b)( 2a – 3b + 1)C) (2a +3b) (2a + 3b – 1) D) None of the above

Simplify algebraic fractions with monomial denominators

We have to simplify fractions to its simplest form (equivalent fractions):

This means the numerator and denominator have no common factor.

Remember the rules for dividing with exponents.

2

3 3 2

15 15 3 5 and

27 27 3 912a 12 6 2

18a 18 6 3a a

mm n

n

aa

a

3 3 13 ( )

6 ( ) 2

x a b x

x b a

Worked Examples:Remember:

(x + 3y) = (3y + x)

-4 x+3y1. 2

2 3y+x

3 23 a - 62. 3

a - 2 2

a

a

Factorise by taking the common factor out and then simplify

If we simplify fractions, we always factorise before cancelling

2 3 2 26

3 9 3 3 3

x x xx x

x x

3.

Test your knowledgeQuestion 7

Simplify: 2y2 – 7xy + 3x2

2x2 - 4xy

Answer

A) 2y – x 2x

C) y+3x -2x

B) (x - y)(y-3x) 2x

D) -(y-3x) 2x