lec 1-38.ppt

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232 Mathematics

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A1.1 Writing expressions

Using symbols for unknowns

+ 9 = 17

Look at this problem:

The symbol stands for an unknown number. We can work out the value of .

= 8

because 8 + 9 = 17

–

= 5

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Using letter symbols for unknowns

In algebra, we use letter symbols to stand for numbers.

These letters are called unknowns or variables.

Sometimes we can work out the value of the letters and sometimes we can’t.

For example,

We can write an unknown number with 3 added on to it as

n + 3

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Writing an expression

Suppose Jon has a packet of biscuits and he doesn’t know how many biscuits it contains.

He can call the number of biscuits in the full packet, b.

If he opens the packet and eats 4 biscuits, he can write an

expression for the number of biscuits remaining in the packet as:

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Writing an equation

Jon counts the number of biscuits in the packet after he has eaten 4 of them. There are 22.

He can write this as an equation:

b – 4 = 22

We can work out the value of the letter b.

b = 26

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Writing expressions

When we write expressions in algebra we don’t usually use the multiplication symbol ×.

For example,

5 × n or n × 5 is written as 5n.

The number must be written before the letter.

When we multiply a letter symbol by 1, we don’t have to write the 1.

For example,

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Writing expressions

When we write expressions in algebra we don’t usually use the division symbol ÷. Instead we use a dividing line as in fraction notation.

For example,

When we multiply a letter symbol by itself, we use index notation.

For example,

n ÷ 3 is written as n 3

n × n is written as n2.

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Writing expressions

Here are some examples of algebraic expressions: n + 7 a number n plus 7

5 – n 5 minus a number n

2n 2 lots of the number n or 2 × n 6

n 6 divided by a number n 4n + 5 4 lots of a number n plus 5

n3 a number n multiplied by itself twice or

n × n × n 3 × (n + 4)

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Identities

When two expressions are equivalent we can link them with the  sign.

For example,

x + x + x  3x

x + x + x is identically equal to 3x

This is called an identity.

In an identity, the expressions on each side of the equation are equal for all values of the unknown.

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A1.2 Collecting like terms

Like terms

An algebraic expression is made up of terms and operators such as +, –, ×, ÷ and ( ).

A term is made up of numbers and letter symbols but not operators.

For example,

3a + 4b – a + 5 is an expression.

3a, 4b, a and 5 are terms in the expression.

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Collecting together like terms

In algebra,

a + a + a + a = 4a

The a’s are like terms.

We collect together like terms to simplify the expression.

 7b + 3b = 10b

 x + 6x – 3x = 4x

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Collecting together like terms

When we add or subtract like terms in an expression we say we are simplifying an expression by collecting

together like terms.

An expression can contain different like terms.

For example,

3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b

= 7a + 8b

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Simplify these expressions by collecting together like terms.

1) a + a + a + a + a = 5a

2) 5b – 4b = b

3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6

= 2c + 2d + 9

4) 4n + n2 – 3n = 4n – 3n + n2 =

5) 4r + 6s – t Cannot be simplified

Collecting together like terms

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Algebraic perimeters

Write an algebraic expression for the perimeter of the following shapes:

2a

3b Perimeter = 2a + 3b + 2a + 3b = 4a + 6b

5x 4y

5x

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A1.3 Multiplying terms

Multiplying terms together

In algebra we usually leave out the multiplication sign ×.

Any numbers must be written at the front and all letters should be written in alphabetical order.

For example,

4 × a = 4a

1 × b = b We don’t need to write a 1 in front of the letter. b × 5 = 5b We don’t write b5.

3 × d × c = 3cd 6 × e × e = 6e2

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Using index notation

Simplify:

x + x + x + x + x = 5x

Simplify:

x × x × x × x × x = x5

x to the power of 5

This is called index notation.

Similarly, x × x = x2

x × x × x = x3

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Exponents



5

3

Power

base

exponent

3

means that is the exponential

3

form of t

Example:

he number

125 5

5

.

125

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The Laws of Exponents:

#1: Exponential form:

The exponent of a power indicates how many times the base multiplies itself.

n

n times

x

x x x

x x x x



      







3

Example: 5

  

5 5 5

n factors of x

#2: Multiplying Powers:

If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS!

2 2

2

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Look at this algebraic expression:

4(a + b) What do do think it means?

Remember, in algebra we do not write the multiplication sign, ×.

This expression actually means:

4 × (a + b) or

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

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Expanding brackets then simplifying

Sometimes we need to multiply out brackets and then simplify.

For example, 3x + 2(5 – x)

We need to multiply the bracket by 2 and collect together like terms.

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Expanding brackets then simplifying

Simplify

4 – (5n – 3)

We need to multiply the bracket by –1 and collect together like terms.

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Expanding brackets then simplifying

Simplify

2(3n – 4) + 3(3n + 5)

We need to multiply out both brackets and collect together like terms.

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 Simplify

5(3a + 2b) – 2(2a + 5b)

We need to multiply out both brackets and collect together like terms.

15a + 10b – 4a –10b = 15a – 4a + 10b – 10b = 11a

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#3: Dividing Powers:

When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS!

m

n

m n

n

x









So, I get it! When you divide Powers, you subtract the exponents!

16

2

₆ ₂ ₄

2 6

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Try these:





2 2

3

.

1





4 2

5

.

2





2 5

.

3

a





7

2

4

2

.

4

s









3

)

2

(

3

)

3

(

.

5





7 3 4

2

.

6

s

t

s

t

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



2 2

3

.

1





4 2

5

.

2





2 5

.

3

a





7

2

4

2

.

4

s









3

)

2

(

3

)

3

(

.

5

81

3

22



4



7 2

5

a





9 7

2

8

4

2



s





s

SOLUTIONS

6 4

2

5





243

)

3

(

)

3

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

4 12

.

7

s



5 9

3

.

8



4 4 8 12

.

9

t

s

t

s



5 4 8 5

4

36

.

10

b

a

b

a

SOLUTIONS 8 4 12

s





81

3

95



4



4 8 4

8 4

12

t

s

t

s

 



3 5 8 4 5

9

4

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#4: Power of a Power:

If you are raising a Power to an exponent, you multiply the exponents!

 

m

n

mn

x



x

6

2

3

2

3

)

5

(







#5: Product Law of Exponents





n

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Try these: home work1

 

3

2 5



.

1

 

3 4



.

2

a

 

2

2 3



.

3

a



2

2 5 3



2



.

4

a

b





3

2

)

2

(

.

5

a

 

2 4 3



.

6

s

t

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#7: Negative Law of Exponents:

1

m m

x





9 3 3 1 125 1 5 1 5 2 2 3 3       and

#8: Zero Law of Exponents:

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Try these: home work 2



2



0



.

1

a

b





4 2

.

2

y

 

5 1



.

3

a





2

4

7

.

4

s



3

2 3



4



.

5

x

y

 

2 4 0



.

6

s

t















2

2 1

.

7

x















2 5 9

3

.

8















2

4 4 2 2

.

9

t

s

t

s















2

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A1.5 Factorizing expressions

Common Factors

)

1

(

5

x

+

10

=

5

)

x

+

2

(

)

2

(

6

a

+ 8

=

(

3

a

+ 4

)

2

(3) 12 – 9

n

=

(

3

n

–

4

)

3

(4) 3

x

+

x

2

=

x

(3 +

x

)

(5) 2

p

+ 6

p

2

– 4

p

3

=

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•

Difference of squares

• a

2

- b

2

= (a - b)(a + b)

or

• a

2

- b

2

= (a + b)(a - b)

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•

Factoring Sums and

Differences of Cubes

a3 + b3 = (a + b)(a2 – ab + b2)

a3_–_b3_{= (}_a_–_b₎₍_a2₊_ab₊_b2₎

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A1.6 Substitution

Substitution

What does

substitution

mean?

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How can be written as an algebraic expression?4 + 3 ×

Using n for the variable we can write this as 4 + 3n

We can evaluate the expression 4 + 3n by substituting different values for n.

When n = 5 4 + 3n = 4 + 3 × 5

= 4 + 15 = 19

When n = 11 4 + 3n = 4 + 3 × 11

= 4 + 33

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can be written as 7n

2

We can evaluate the expression by substituting different values for n.

7n 2

When n = 4 7n

2 = 7 × 4 ÷ 2 = 28 ÷ 2 = 14

When n = 1.1 7n

2 = 7 × 1.1 ÷ 2 = 7.7 ÷ 2

7 ×

2

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can be written as n2 + 6

We can evaluate the expression n2 + 6 by substituting

different values for n.

When n = 4 n2 + 6 = 42 + 6

= 16 + 6 = 22

When n = 0.6 n2 + 7 = 0.62 + 6

= 0.36 + 6

2 + 6

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can be written as 2(n + 8)

We can evaluate the expression 2(n + 8) by substituting

different values for n.

When n = 6 2(n + 8) = 2 × (6 + 8) = 2 × 14

= 28

When n = 13 2(n + 8) = 2 × (13 + 8) = 2 × 21

2( + 8)

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Here are five expressions.

1) a + b + c

2) 3a + 2c

3) a(b + c)

4) abc

5)

a b2 – _c

Evaluate these expressions when a = 5, b = 2 and c = –1

= 5 + 2 + –1 = 6

= 3 × 5 + 2 × –1 = 15 + –2 = 13 = 5 × (2 + –1) = 5 × 1 = 5

= 5 × 2 × –1= 10 × –1 = –10 = 5 ÷ 5 = 1

5 22 – –1

=

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lec 1-38.ppt

232 Mathematics

A1.1 Writing expressions

Contents

A1.2 Collecting like terms

Using symbols for unknowns

Writing an expression

Writing an equation

Writing expressions

Identities

A1.2 Collecting like terms

Contents

A1.1 Writing expressions

Like terms

Collecting together like terms

Algebraic perimeters

A1.3 Multiplying terms

Contents

A1.1 Writing expressions

Multiplying terms together

Exponents

Expanding brackets then simplifying

Expanding brackets then simplifying

SOLUTIONS

A1.5 Factorizing expressions

Contents

A1.1 Writing expressions

Common Factors

A1.6 Substitution

Contents

A1.1 Writing expressions

Substitution

substitution