Algebraic Expressions

Basic Definitions A term is a single item such as: An expression is a collection of terms d 5b -2c 3c2c3d2a 2a+3a3b-b4g-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. 3d5d 3c2c ++ = = 3d5d 3c2c 8d 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 + 4y7a + 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 = 5 3 Exponentials can also be expressed in algebraic form as well: Y x Y x Y x Y x Y = y 5

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 right Instructions: 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 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 (3a+2) = 2 (3a+2) =6a Some Practice Questions

3(2b+1) 3 (2b+1) = 3 (2b+1) =6b +3 +3

5(4t+5s) 5 (4t+5s) = 20t 20t +25s +25s

3(2d-3e) 3 (2d-3e) = 3 (2d-3e) = 6d 6d -9e -9e

7a(2b-3c) 7a (2b-3c) = 14ab-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 5 y = xy + 5x + 5y + 25 XY 5X 5Y 25 There 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 y = ay - 6a + 5y - 30 ay -6a 5y -30 There 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 a = a 2 - 4a + 10a - 40 a2a2 -4a 10a -40 There are LIKE TERMS so we need to simplify the expression = a 2 + 6a - 40

Perfect square rule

Perfect Squares Rule Use when the sign is positiveUse when the sign is negative

Difference of two squares rule for multiplication 101× 99 = (100 +1)(100 −1) = – = −1 2 = −1 = × 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) = a 2 -ab+ab-b 2 = a 2 -b 2 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 Expressions 12y Factorise the expression: 12y + 24 Highest Common Factors: Number PartPronumeral Part y y Highest 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 (y + 2)

Factoring: Algebraic Expressions 14a Factorise the expression: 14a Highest Common Factors: Number PartPronumeral Part a a Highest 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 – 5 7 (2a – 5)

Factoring: Algebraic Expressions 24abc Factorise the expression: 24abc – 10b Highest Common Factors: Number PartPronumeral Part abc b ac -10b Highest Common Factors: In this example the common factors for both terms are 2 and b therefore the HCF is 2b = 2 x b Therefore we divide the original expression by 2b We then represent it in factorised form: (24abc – 10b) ÷ 2b = 12ac - 5 2b (12ac - 5) b b 1 Pronumeral PartNumber Part 2 2

Grouping ‘two by two’

ax 2 +bx+cx+3x Original 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:

Examples:

Factorising Perfect Squares

Step by Step 4x x + 25 Therefore 4x x + 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, 25x 2 = (5x) 2 Yes, 9 = 3 2 Determine whether is a perfect square trinomial. If so, factor it. 25x x + 9 Answer:is a perfect square trinomial. 25x x + 9 Example 1

We Know thatis a perfect square trinomial. 25x x + 9 Remember the perfect squares rule: a 2 + 2ab + b 2 = (a + b) 2 a 2 - 2ab + b 2 = (a – b) 2 Factorising a perfect square trinomial But how do we factorise it? 25x x = (3) 2 Therefore b = 3 25x 2 = (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, 49y 2 = (7y) 2 Yes, 36 = 6 2 Determine whether is a perfect square trinomial. If so, factor it. 49y y + 36 Answer:is not a perfect square trinomial. 49y y + 36 Example 2

Factorising using the difference of two Squares a 2 - b 2 = (a + b)(a - b)

Difference of Squares a 2 - b 2 = (a - b)(a + b) or a 2 - b 2 = (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!!!

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 = 5 2 = 5 x 5 Determine whether is a perfect square binomial. If so, factor it. Yes, X Example 1 x Are there only 2 terms? Yes, x Yes, X 2 = X x X Lets Factor it : x 2 – 25 ( )( ) 5xx+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 = 3 2 = 3 x 3 Determine whether is a perfect square binomial. If so, factor it. Yes, 16X Example 2 16x Are there only 2 terms? Yes, 16x Yes, 16X 2 = 4X x 4X Lets Factor it : 16x 2 – 9 ( )( ) 34x +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 Ax 2 + Bx + C (x + 3) (x + 4) = x 2 + 4x + 3x + 12 Gives us a Quadratic Trinomial = x 2 + 7x + 12

The Pattern (x + 3) (x + 4) = x 2 + 4x + 3x + 12 = x 2 + 7x + 12 Ax 2 + Bx + C x 2 + 7x + 12 The 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: x 2 + 8x Place the X values in brackets (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) = x 2 + 5x + 3x + 15 = x 2 + 8x + 15