Algebraic Expressions

Basic DefinitionsA term is a single item such as:

An expression is a collection of terms

d 5b-2c

3c 2c 3d

2a

2a+3a 3b-b 4g-2g+g

Expanding on the definition

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or - signs)

Like Terms

"Like terms" are terms whose variables are the same.

In an expression, only like terms can be combined.

3d 5d3c 2c

+ += =3d 5d 3c 2c8d 5c

Simplifying Expressions Expressions can be ‘simplified’ by collecting

like terms together. Simple expressions:

Complex Expressions:

2a+3a 3b-b4g-2g+g= = =

5a + 3y + 3a + 4y 7a + 6y + 3a + 7y= =

5a 3g 2b

8a + 7y 10a + 13y

But what about exponentials?Remember: Exponents are shorthand for repeated

multiplication of the same thing by itself. For example:

5 x 5 x 5 = 53 Exponentials can also be expressed in algebraic form as well:

Y x Y x Y x Y x Y = y5

Expanding Brackets a(b+c)

The Frog Puzzle

The objective is to get all three frogs on each side across to the opposite side, such that, the green frogs are lined up on the left side lily pads, and the blue frogs end up on the rightInstructions:

What is the smallest amount of moves you need to complete this puzzle ?

Try it out for yourself!

Draw a series of boxes like this in your book

Leave the middle square empty

Collect 2 lots of 5 counters that are the same colour

Try solving the puzzle with:• 3 Counters on each side • 4 Counters on each side • 5 Counters on each side

Record your smallest amount of moves for each into your books !

Lets look at the Pattern Number of Frogs

on Each Side = N

Number of Hops

Number of Slides

Minimum number of moves

1 1 2 32 4 4 83 9 6 154 16 8 245 25 10 35

Look at the first and last column can you see a pattern?

N (N+2)

Can you create an algebraic expression of the form a(b+c) that will fit the data

Problem: 8 frogs! Using the equation below:

Can you figure out the minimum number of moves needed for eight red frogs to change places with eight green frogs ?

N (N+2)

2(3a+2)

2 (2 (3a3a+2+2) ) = =

6a6a +4+4

Some Practice Questions

3(2b+1)

3 (3 (2b2b+1+1) ) = =

6b6b +3+3

5(4t+5s)

5 (5 (4t4t+5s+5s) ) = =

20t20t +25+25ss

3(2d-3e)

3 (3 (2d2d-3e-3e) ) ==

6d6d -9e-9e

7a(2b-3c)

7a (27a (2bb-3c-3c) ) ==

14ab14ab-21ac-21ac

Alternative Method: Boxes What is 2(3x + 4)?

Expanding Brackets (a+b)(c+d)

Expanding Double Brackets

(a+b)(c+d)

= ac + ad + bc + bd

When expanding double brackets we

can simply draw arrows to indicate

each term to multiply

Factorised Form

Expanded Form

However this method can seem confusing so we will be using the box method

Box Method: Example 1 Lets expand (x+5) (y+5) using the box method

X

5

5y

= xy + 5x + 5y + 25 XY 5X

5Y 25There are NO LIKE TERMS so we don’t need to do anything

else

Box Method: Example 2 Lets expand (a+5) (y-6) using the box method

a

5

- 6y

= ay - 6a + 5y - 30ay -6a

5y -30There are NO LIKE TERMS so we don’t need to do anything

else

Box Method: Example 3 Lets expand (a+10) (a-4) using the box method

a

10

- 4a = a2 - 4a + 10a - 40

a2 -4a

10a -40

There are LIKE TERMS so we need to simplify the

expression

= a2 + 6a - 40

Perfect square rule

Perfect Squares RuleUse when the sign is positive Use when the sign is negative

Difference of two squares rule for multiplication

101× 99 = (100 +1)(100 −1)= 1002 – 100 +100 -12

= 1002 −12

= 10000 −1= 9999

101× 99 = ?How could you solve the following without using a calculator?

We can use the difference of two squares to solve this

= (a+b) (a-b)= a2-ab+ab-b2

= a2 -b2

Worked Example: Formula Example:

Factorising Using Common Factors

Factorising Previously we have been EXPANDING terms (i.e.

removing the brackets) We will now begin to FACTORISE terms (i.e. with

brackets)

But before we begin factoring algebraic expressions, Lets review how to factor simple numbers

7( a + 2) 7a + 14 Factorised Form Expanded Form

= 7 x a + 7 x 2

Factor Trees Original Number

Factors of 36

Factors of 9 and 4

-Prime Number(Only divisible by itself or 1)

Factor OF (non-prime number, can be further divided)

Another Example: Factors of 48

-Prime Number(Only divisible by itself or 1)

Factor OF (non-prime number, can be further divided)

Activity: Practice Questions

Now lets try to find the HIGHEST

COMMON FACTOR of 2

simple numbers

Factoring: Algebraic Expressions12y

Factorise the expression: 12y + 24

Highest Common Factors:

Number Part Pronumeral Part

y

y 1

+24Highest Common Factors:

In this example the common factors for both terms are 3, 2 and 2 therefore the HCF is 12 = 3 x 2 x 2

Therefore we divide the original expression by 12

We then represent it in factorised form:

(12y + 24) ÷ 12 = y + 2

12 (y + 2)

+24

6 43 2 2 2

12

6 23 2

Factoring: Algebraic Expressions

14aFactorise the expression: 14a - 35

- 35

7 5

14

7 2

Highest Common Factors:

Number Part Pronumeral Part

a

a 1

-35Highest Common Factors:

In this example the only common factor is 7

Therefore we divide the original expression by 7

We then represent it in factorised form:

(14a – 35) ÷ 7 = 2a – 57 (2a – 5)

Factoring: Algebraic Expressions24abc

Factorise the expression: 24abc – 10b

Highest Common Factors:

Number Part Pronumeral Part

abc

b ac

-10bHighest Common Factors:

In this example the common factors for both terms are 2 and b therefore the HCF is 2b = 2 x bTherefore we divide the original expression by 2b

We then represent it in factorised form:

(24abc – 10b) ÷ 2b = 12ac - 5

2b (12ac - 5)

-10

5 2

24

6 43 2

b

b 1

Pronumeral PartNumber Part

2 2

Grouping ‘two by two’

ax2+bx+cx+3xOriginal Expression

X is the only common factor and is removed

x(ax+b+c+3)

7x + 14y + bx + 2by Original Expression

Simple Example:

Common factor of 7

Common factor of b

= (7x + 14y) + (bx + 2by)= 7(x+2y) +b(x+2y)

= (x+2y)(7+b)

Grouping ‘Two by Two’ Example:

Step One: Look for common factors.

Step Two: group factors by common factors.

Step Three: take out the common factor in each pair.

Step four: Remove common factor in the brackets

7x + 14y + bx + 2by Original Expression

Common factor of 7

Common factor of b

= (7x + 14y) + (bx + 2by)

= 7(x+2y) +b(x+2y)

= (x+2y)(7+b)

Grouping ‘Two by Two’ Example:

1

2

3

4

Examples:

Factorising Perfect Squares

Step by Step

4x2 + 20x + 25

Therefore 4x2 + 20x + 25 is a perfect square trinomial

3. Is the middle term equal to ?2(5x)(3) Yes 30x = 2(5x)(3)

1. Is the first term a perfect square?

2. Is the last term a perfect square?

Yes, 25x2 = (5x)2

Yes, 9 = 32

Determine whether is a perfect square trinomial. If so, factor it.

25x2 + 30x + 9

Answer: is a perfect square trinomial.25x2 + 30x + 9

Example 1

We Know that is a perfect square trinomial.

25x2 + 30x + 9

Remember the perfect squares rule:

a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a – b)2

Factorising a perfect square trinomial

But how do we factorise it?25x2 + 30x + 9

9 = (3)2

Therefore b = 3 25x2 = (5x)2

Therefore a = 5x

Answer:(5x + 3)

3. Is the middle term equal to ?2(7y)(6) No, 42y ≠ 2(7y)(6) = 84y

1. Is the first term a perfect square?

2. Is the last term a perfect square?

Yes, 49y2 = (7y)2

Yes, 36 = 62

Determine whether is a perfect square trinomial. If so, factor it.

49y2 + 42y + 36

Answer: is not a perfect square trinomial.49y2 + 42y + 36

Example 2

Factorising using the difference of two Squares

a2 - b2 = (a + b)(a - b)

Difference of Squaresa2 - b2 = (a - b)(a + b)

or

a2 - b2 = (a + b)(a - b)

The order does not matter!!

4 Steps for factoring Difference of Squares

Are there only 2 terms?

Is the first term a perfect square?

Is the last term a perfect square?

Is there subtraction (difference) in the problem?

If all of these are true, you can factor using this method!!!

1

23

4

4. Is there a subtraction in the expression?

2. Is the first term a perfect square?

3. Is the last term a perfect square? Yes, 25 = 52 = 5 x 5

Determine whether is a perfect square binomial. If so, factor it.

Yes, X2 - 25

Example 1x2 - 25

1. Are there only 2 terms? Yes, x2 - 25Yes, X2 = X x X

Lets Factor it :

x2 – 25

( )( )5 xx + 5-

4. Is there a subtraction in the expression?

2. Is the first term a perfect square?

3. Is the last term a perfect square? Yes, 9 = 32 = 3 x 3

Determine whether is a perfect square binomial. If so, factor it.

Yes, 16X2 - 9

Example 216x2 - 9

1. Are there only 2 terms? Yes, 16x2 - 9Yes, 16X2 = 4X x 4X

Lets Factor it :

16x2 – 9

( )( )3 4x4x+ 3-

Factorising Quadratic Trinomials

What is a Quadratic trinomial?Expanding 2 factors such as:

A Quadratic Trinomial has two important features: • The highest power of a pronumeral is 2 • There are three terms present

Ax2 + Bx + C

(x + 3) (x + 4)= x2 + 4x + 3x + 12

Gives us a Quadratic Trinomial= x2 + 7x + 12

The Pattern(x + 3) (x + 4)

= x2 + 4x + 3x + 12= x2 + 7x + 12

Ax2 + Bx + C

x2 + 7x + 12The numbers 3 & 4 multiply to give 12 or the C

term

Both numbers also add to give us the 7x or the B term

The A terms are a result of the multiplication of the X pronumeral

Lets try another one:

x2 + 8x + 15

1 Place the X values in brackets (x ) (x )

2 What two numbers must multiply to give 15 but add to give 8 (x + 3) (x + 5)

3 Check you expression by expanding it

(x + 3) (x + 5)= x2 + 5x + 3x + 15

= x2 + 8x + 15