Grade 9 Mathematics: Algebraic expressions

Grade 9 Mathematics: Algebraic expressions

Fill in the following definitions: Expression- __________________________________________________________ Term- ______________________________________________________________ Coefficient- __________________________________________________________ Constant- ____________________________________________________________ Degree of the expression- _______________________________________________ Like terms- __________________________________________________________ Distribute- ___________________________________________________________ Factorise- ____________________________________________________________ Simplify- ____________________________________________________________

Go through the Platinum textbook p.70 for additional notes on conventions in expressions.

Exercise 1 1. Consider the expression 43 - 52 + + 3

a) How many terms are there in the expression? b) What is the constant term? c) What is the coefficient of 2? d) What is the degree of the expression? e) What is the value of the expression if = 3?

Remember you may only add or subtract like terms e.g. 22 + 32 = 52 but 22 + 32 522

2. Simplify the following: a) 43 - 52 + + 2 - 33 + 2 + 3 - 3 + b) 22 - 52 + 2 - 3 + 52 c) (43 - 52 + + 3) - (63 - 32 + 5 + 9) d) Subtract (-32 + 7 + 4) from (2 + 3 + 2)

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Grade 9 Mathematics: Algebraic expressions

Multiplication and division

Numbers outside the brackets must be multiplied into the brackets (distributed).

e.g. 2(3 + 4)

2 ? (3 + 4)

= 2 ? 3 + 2 ? 4

= 6 + 8

Note: The result would be the same if the expression was (3 + 4)2

Multiplication of two binomials

e.g. (5 - 2)(4 + 3)

= 5(4 + 3) - 2(4 + 3) = 202 + 15 - 8 - 6 = 202 + 7 - 6

,

The same result can be achieved by using the acronym FOIL

This when you take the First, Outer, Inner and Last terms from the brackets and multiply them together respectively.

e.g. (5 - 2)(4 + 3)

= (5 ? 4) + (5 ? 3) + (-2 ? 4) + (-2 ? 3) = 202 + 15 - 8 -6 = 202 + 7 - 6

Division by a monomial

e.g. 93+32-

3

= 93 + 32 -

3 3 3

= 3 + - 1

3

Remember to apply the laws of exponents when multiplying or dividing in expressions. Also remember to follow the correct order when dealing with multiple operations.

e.g. 3 + 2( + 4) = 3 + 22 + 8 = 22 + 11

3 2 2

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Grade 9 Mathematics: Algebraic expressions

Exercise 2: Simplify

1. 2 - 2

3. (33)2 + 162 - 84 5. 3(2 - 5) + 2(3 - 2)

7. -2(3 - 4)

9. ( + 3)(2 - 3)

11. (42 + 2)(4 - 3 + 5)

13.

152+323-27 3

15. Given = -1; = 2; = 0 evaluate:

2. 3 - 5 +

4. 4(2 - 3)

6. (2 + 4 - 3)

8. (2 - 5)(2 + 5)

10. (3 + 5)2

12. 1 (142 + 83)

2

14. (22 + 83) ? 4

5 + 2 - 2

Factorisation

Factorisation is the opposite of distribution, you need to find out what are the factors of the expression. For example the factors of 22 + 6 are 2 and ( + 3) because you have to multiply 2( + 3) to get 22 + 6.

The types of factorisation that will be dealt with in grade 9 are: Highest Common Factor (HCF), Grouping (a form of HCF), Difference Of Two Squares, and Trinomial.

The following are all examples where you have to factorise.

Highest common factor e.g. 213 - 152 + 3

= 3(72 - 5 + 1)

, 3 left over"

The terms in the bracket come from dividing the terms in the original expression by the highest common factor that was found i.e. 213 ? 3 = 72; -152 ? 3 = 5 3 ? 3 = 1.

Note: the number and/ or variable outside the bracket must always be the highest common factor not just any common factor; and the number of terms inside the bracket should be the same as the number of terms in the original expression.

Grouping

Sometimes some terms have something in common and other terms have something else in common, this is where grouping is used.

e.g. 2 + 4 - - 2 = 2( + 2) - ( + 2)

( + 2)

= ( + 2)(2 - )

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Grade 9 Mathematics: Algebraic expressions

Difference Of Two Squares

This type of factorisation can only be used when the following three conditions are met. The expression must only have two terms, the terms must be separated by a minus sign and all the numbers and variables must be square numbers and variables.

e.g. 162 - 94

= (4 - 32)(4 + 32)

The form of these answers are two brackets each containing two terms.

The numbers and variables in the brackets must be the same but the signs must be different i.e. one bracket has a plus and one has a minus but they both have 4 32.

It does not matter if you put the plus in the first bracket and minus in the second or the other way around.

The way to determine which numbers and variables go inside the bracket is by finding the square roots of the terms in the original expression i.e. 162 = 4 94 = 32.

Trinomial

This type of factorisation can only be used for expression with the form 2 + + where , are constants (numbers).

e.g. 2 + 8 + 12

= 1, = 8 = 12

= ( + 6)( + 2)

The form of these answers are also two binomials.

The variables and numbers inside the brackets come from the factors of the first and last terms in the original expression. i.e. 2 = ? 12 = 6 ? 2.

Note: the pairs of factors we can use to get 12 are 12 ? 1 6 ? 2 4 ? 3 but we have to use 6 and 2 because they can add to give the coefficient of the second term (8) in the original expression.

More examples

a) 2 + 2 - 15 = ( + 5)( - 3)

5 ? (-3) = -15 5 + (-3) = 2

c) 22 + 15 - 8 = (2 - 1)( + 8)

2 ? 8 = 16 (-1) ? = - 16 + (-) = 15

b) 2 - 9 + 20 = ( - 5)( - 4)

(-5) ? (-4) = 20 (-5) + (-4) = -9

2 ? = 22; (-1) ? 8 = -8 and we have to consider how the factors have to work together to get the second term from the question. You need to figure out which factors to use if there is more than one pair and figure out in which bracket to put each number. This is because the coefficient of 2 1

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Grade 9 Mathematics: Algebraic expressions

Note: always check is terms have common factors and take out the HCF first, then do other types of factorisation if possible.

Factorisation answers can always be checked by multiplying out the factors and checking that it is the same as the original question that was given.

e.g. factorise 43 + 122

check: 42 ? + 42 ? 3

= 42( + 3)

= 43 + 122

Exercise 3

1. Factorise by taking out the highest common factor.

a) -9 + 3 - 6

b) 122 - 602 + 4822

c) 12 - 15 + 162 - 20

d) 8(3 - ) + 7( - 3)

2. Factorise using the difference of two squares method.

a) 92 - 642

b) 164 - 1

c) 81 - 1216

d) (3 + )2 - 2

3. Factorise using the trinomial method.

a) 2 + 14 + 40 c) 2 - 12 + 36

b) 2 - - 20 d) 32 + 11 + 10

4. Factorise fully (use any method necessary)

a) 2 - 6 - 7

b) 32 - 12

c) 752 + 25

d) 32 - 3 - 60

e)

2 + 5 + 1

6

6

f) 42 - 368

5. Simplify these algebraic fractions. Remember to factorise where possible.

e.g.

2-16 2+6+8

= (-4)(+4)

(+4)(+2)

( + 4) ? ( + 4) = 1 ""

= -4

+2

a)

4+2 2-1

c)

3+24 2-16

?

2-36 2+14+48

e)

2+8+15 32+9

?

2-25 62+42

b)

2+9+14 2+11+28

d)

2+3+2 8+8

?

9+27 2-9

f)

2-9 3+15

?

2+3-10 2+2+1

?

(-3)(-2) 3(+3)(+1)2

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