Transcript Algebraic Expressions (BD)

Algebraic Expressions
Basic Definitions
A term is a single item such as:
2a
-2c
d
3c
2c
5b
3d
An expression is a collection of terms
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.
3d

3c
5d
2c
In an expression, only like terms can be
combined.
3d
+
5d
= 8d
3c
+
2c
= 5c
Simplifying Expressions

Expressions can be ‘simplified’ by collecting
like terms together.
Simple expressions:
2a+3a = 5a
4g-2g+g = 3g
3b-b = 2b
Complex Expressions:
5a + 3y + 3a + 4y
= 8a + 7y
7a + 6y + 3a + 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 right
Instructions:
What is the smallest amount of moves you need to
complete this puzzle ?
Try it out for yourself!
Collect 2 lots of 5 counters that are the same colour
Draw a series of boxes like this in your book
Leave the middle square empty
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
2
1
4
2
4
3
8
3
4
5
9
16
25
6
8
10
15
24
35
Look at the first and last column can you see a pattern?
Can you create an algebraic expression of the form
a(b+c) that will fit the data
N (N+2)
Problem: 8 frogs!

Using the equation below:
N (N+2)

Can you figure out the minimum number of
moves needed for eight red frogs to change
places with eight green frogs ?
Some Practice Questions
 2(3a+2)
2 (3a+2) = 6a +4
 3(2b+1)
3 (2b+1) = 6b +3
 5(4t+5s)
5 (4t+5s) = 20t +25s
 3(2d-3e)
3 (2d-3e) = 6d -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
Factorised
Form
Expanded
Form
(a+b)(c+d)
When expanding
double brackets we
can simply draw
arrows to indicate each
term to multiply
= ac + ad + bc + bd
However this method can seem confusing so we will be using the box method
Box Method: Example 1

X
Lets expand (x+5) (y+5) using the box
method
y
5
XY
5X
= xy + 5x + 5y + 25
5
5Y
25
There are NO LIKE TERMS so
we don’t need to do anything
else
Box Method: Example 2

a
Lets expand (a+5) (y-6) using the box
method
y
-6
ay
-6a
= ay - 6a + 5y - 30
5
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
10
a2
10a
-4
-4a
-40
= a2 - 4a + 10a - 40
There are LIKE TERMS so we
need to simplify the
expression
= a2 + 6a - 40
Perfect square rule
Perfect Squares Rule
Use when the sign is positive
Use when the sign is negative
Difference of two squares rule for
multiplication
How could you solve the following without using a calculator?
101× 99 = ?
We can use the difference of two squares to solve this
Worked Example:
101× 99 = (100 +1)(100 −1)
= 1002 – 100 +100 -12
= 1002 −12
= 10000 −1
= 9999
Formula Example:
= (a+b) (a-b)
= a2-ab+ab-b2
= a2 -b2
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)


7( a + 2)
7a + 14
Factorised Form
Expanded Form
=7xa+7x2
But before we begin factoring algebraic
expressions, Lets review how to factor
simple numbers
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
Factorise the expression: 12y + 24
12y
Highest Common Factors:
Number Part
Pronumeral Part
12
y
6
3
2
y
+24
Highest Common Factors:
+24
6
1
2
3
4
2 2 2
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: 12 (y + 2)
(12y + 24) ÷ 12 = y + 2
Factoring: Algebraic Expressions
Factorise the expression: 14a - 35
14a
-35
Highest Common Factors:
Number Part
Pronumeral Part
14
a
7
2
a
Highest Common Factors:
- 35
1
7
5
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
Factorise the expression: 24abc – 10b
24abc
Highest Common Factors:
Number Part
Pronumeral Part
24
6
3
4
2 2 2
abc
b
ac
-10b
Highest Common Factors:
Number Part
Pronumeral Part
b
-10
5
2
b
1
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
(24abc – 10b) ÷ 2b = 12ac - 5
We then represent it in factorised form:
2b (12ac - 5)
Grouping ‘two by two’
Simple Example:
ax2+bx+cx+3x
Original Expression
Grouping ‘Two by Two’ Example:
7x + 14y + bx + 2by
Original Expression
X is the only common
factor and is removed
x(ax+b+c+3)
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:
7x + 14y + bx + 2by
1
Original Expression

Common
factor of b
Common
factor of 7


= (7x + 14y) + (bx + 2by)
2
= 7(x+2y) +b(x+2y)
3
= (x+2y)(7+b)
4

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
Examples:
Factorising Perfect
Squares
Step by Step
4x2 + 20x + 25
Therefore
4x2 + 20x + 25 is a perfect square trinomial
Example 1
Determine whether 25x2 + 30x + 9
square trinomial. If so, factor it.
is a perfect
1. Is the first term a perfect square?
Yes, 25x2 = (5x)2
2. Is the last term a perfect square?
Yes,
9 = 32
3. Is the middle term equal to 2(5x)(3) ? Yes 30x = 2(5x)(3)
Answer: 25x2 + 30x + 9 is a perfect square trinomial.
Factorising a perfect square trinomial
We Know that 25x2 + 30x + 9 is 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)
Remember the perfect squares rule:
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a – b)2
Example 2
Determine whether 49y2 + 42y + 36 is a perfect
square trinomial. If so, factor it.
1. Is the first term a perfect square?
Yes, 49y2 = (7y)2
2. Is the last term a perfect square?
Yes,
36 = 62
3. Is the middle term equal to 2(7y)(6) ? No, 42y ≠ 2(7y)(6) = 84y
Answer: 49y2 + 42y + 36 is not a perfect square trinomial.
Factorising using the
difference of two
Squares
2
a
-
2
b
= (a + b)(a - b)
Difference of Squares
a2 - b2 = (a - b)(a + b)
or
a2 - b2 = (a + b)(a - b)
The order does not matter!!
4 Steps for factoring Difference of Squares
1
Are there only 2 terms?
2
Is the first term a perfect square?
3
Is the last term a perfect square?
4
Is there subtraction (difference) in the problem?
If all of these are true, you can factor using this method!!!
Example 1
x2 - 25
Determine whether
square binomial. If so, factor it.
is a perfect
1. Are there only 2 terms?
2. Is the first term a perfect square?
Yes, x2 - 25
Yes, X2 = X x X
3. Is the last term a perfect square?
Yes,
25 = 52 = 5 x 5
4. Is there a subtraction in the
expression?
Yes,
X2 - 25
x2 – 25
Lets Factor it :
( x + 5 )(x - 5 )
Example 2
16x2 - 9
Determine whether
square binomial. If so, factor it.
is a perfect
1. Are there only 2 terms?
2. Is the first term a perfect square?
Yes, 16x2 - 9
Yes, 16X2 = 4X x 4X
3. Is the last term a perfect square?
Yes,
4. Is there a subtraction in the
expression?
9 = 32 = 3 x 3
Yes, 16X2 - 9
16x2 – 9
Lets Factor it :
( 4x + 3 )(4x - 3 )
Factorising Quadratic
Trinomials
What is a Quadratic trinomial?
Expanding 2 factors such as:
(x + 3) (x + 4)
= x2 + 4x + 3x + 12
Gives us a Quadratic Trinomial
= x2 + 7x + 12
A Quadratic Trinomial has two important features:
• The highest power of a pronumeral is 2
• There are three terms present
Ax2 + Bx + C
The Pattern
(x + 3) (x + 4)
= x2 + 4x + 3x + 12
= x2 + 7x + 12
The A terms are a
result of the
multiplication of
the X pronumeral
x2 + 7x + 12
Both numbers also
add to give us the 7x
or the B term
Ax2 + Bx + C
The numbers 3
& 4 multiply to
give 12 or the C
term
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