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Everything Maths www.everythingmaths.co.za
1. Algebraic expressions
Grade 10
2
Everything Maths www.everythingmaths.co.za
The real number system
The real number system consists of rational and irrational numbers. Rational
numbers include the integers, whole and natural numbers.
● Integers are { . . . ; -3; -2; -1; 0; 1; 2; 3; . . . }
● Whole numbers are {0; 1; 2; 3; . . . }
● Natural numbers are {1; 2; 3; . . . }
3
Everything Maths www.everythingmaths.co.za
Rational numbers
The following are rational numbers:
● Fractions with both numerator and denominator as integers
● Integers
● Decimal numbers that terminate
● Decimal numbers that recur (repeat)
A rational number is any number that can be written as
a
b
where a and b are integers and b≠0
Irrational numbers
Irrational numbers are numbers that cannot be written as a fraction with the numerator
and denominator as integers.
4
Everything Maths www.everythingmaths.co.za
Rounding off
Rounding off a decimal number allows us to approximate a number.
For example to round 2,6525272 to three decimal places:
● count three places after the decimal and place a | between the third and fourth numbers
● round up the third digit if the fourth digit is greater than or equal to 5
● leave the third digit unchanged if the fourth digit is less than 5
● if the third digit is a 9 and needs to be rounded up, then the 9 becomes a 0 and the second
digit is rounded up
So for 2,6525272 we place the marker: 2,652|5272
Then we note that the fourth digit is a 5 so we round up: 2,653
5
Everything Maths www.everythingmaths.co.za
Estimating surds
If n = 2:
● A perfect square is the number obtained when an integer is squared.
● We can use perfect squares to determine between which two integers a square root lies.
If n = 3:
● A perfect cube is the number obtained when an integer is cubed.
● We can use perfect cubes to determine between which two integers a cube root lies.
If a and b are positive whole numbers, and a<b , then
n
√a=
n
√b
2<√7<3
because 2
2
=4 and 3
2
=9
and √4<√7<√9
2<
3
√10<3
because 23
=8 and 33
=27
and
3
√8<
3
√10<
3
√27
6
Everything Maths www.everythingmaths.co.za
Terminology for mathematical expressions
A monomial is an expression with one term.
A binomial is an expression with two terms.
A trinomial is an expression with three terms.
7
Everything Maths www.everythingmaths.co.za
Products
● The product of a monomial with a binomial is ax(cx + d) = acx2
+ adx
● The product of two binomials is (ax + b)(cx + d) = acx2
+ x(ad + bc) + bd
● The product of a binomial and a trinomial is:
(A + B)(C + D + E) = A(C + D + E) + B(C + D + E)
● The product of two identical binomials is known as the square of the binomial.
● We get the difference of two squares when we multiply (ax + b)(ax − b) = (ax)2
- b2
8
Everything Maths www.everythingmaths.co.za
Factorisation
● Factorisation is the opposite process of expanding the brackets.
● Taking out a common factor is the basic factorisation method.
● We often need to use grouping in pairs to factorise polynomials.
● To factorise a quadratic we find the two binomials that were multiplied together to give
the quadratic.
9
Everything Maths www.everythingmaths.co.za
Difference and sum of two cubes
● The sum of two cubes can be factorised as:
x3
+ y3
= (x + y)(x2
− xy + y2
)
● The difference of two cubes can be factorised as:
x3
− y3
= (x − y)(x2
+ xy + y2
)
10
Everything Maths www.everythingmaths.co.za
Simplification of fractions
● We can simplify fractions by incorporating the methods we have learnt to factorise
expressions.
● Only factors can be cancelled out in fractions, never terms.
● To add or subtract fractions, the denominators of all the fractions must be the same.
a
b
×
c
d
=
ac
bd
, (b≠0,d≠0)
a
b
+
c
b
=
a+c
b
, (b≠0)
a
b
÷
c
d
=
a
b
×
d
c
=
ad
bc
, (b≠0,c≠0,d≠0)
11
Everything Maths www.everythingmaths.co.za
For more practice see:
www.everythingmaths.co.za

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Algebraic expressions

  • 1. 1 Everything Maths www.everythingmaths.co.za 1. Algebraic expressions Grade 10
  • 2. 2 Everything Maths www.everythingmaths.co.za The real number system The real number system consists of rational and irrational numbers. Rational numbers include the integers, whole and natural numbers. ● Integers are { . . . ; -3; -2; -1; 0; 1; 2; 3; . . . } ● Whole numbers are {0; 1; 2; 3; . . . } ● Natural numbers are {1; 2; 3; . . . }
  • 3. 3 Everything Maths www.everythingmaths.co.za Rational numbers The following are rational numbers: ● Fractions with both numerator and denominator as integers ● Integers ● Decimal numbers that terminate ● Decimal numbers that recur (repeat) A rational number is any number that can be written as a b where a and b are integers and b≠0 Irrational numbers Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers.
  • 4. 4 Everything Maths www.everythingmaths.co.za Rounding off Rounding off a decimal number allows us to approximate a number. For example to round 2,6525272 to three decimal places: ● count three places after the decimal and place a | between the third and fourth numbers ● round up the third digit if the fourth digit is greater than or equal to 5 ● leave the third digit unchanged if the fourth digit is less than 5 ● if the third digit is a 9 and needs to be rounded up, then the 9 becomes a 0 and the second digit is rounded up So for 2,6525272 we place the marker: 2,652|5272 Then we note that the fourth digit is a 5 so we round up: 2,653
  • 5. 5 Everything Maths www.everythingmaths.co.za Estimating surds If n = 2: ● A perfect square is the number obtained when an integer is squared. ● We can use perfect squares to determine between which two integers a square root lies. If n = 3: ● A perfect cube is the number obtained when an integer is cubed. ● We can use perfect cubes to determine between which two integers a cube root lies. If a and b are positive whole numbers, and a<b , then n √a= n √b 2<√7<3 because 2 2 =4 and 3 2 =9 and √4<√7<√9 2< 3 √10<3 because 23 =8 and 33 =27 and 3 √8< 3 √10< 3 √27
  • 6. 6 Everything Maths www.everythingmaths.co.za Terminology for mathematical expressions A monomial is an expression with one term. A binomial is an expression with two terms. A trinomial is an expression with three terms.
  • 7. 7 Everything Maths www.everythingmaths.co.za Products ● The product of a monomial with a binomial is ax(cx + d) = acx2 + adx ● The product of two binomials is (ax + b)(cx + d) = acx2 + x(ad + bc) + bd ● The product of a binomial and a trinomial is: (A + B)(C + D + E) = A(C + D + E) + B(C + D + E) ● The product of two identical binomials is known as the square of the binomial. ● We get the difference of two squares when we multiply (ax + b)(ax − b) = (ax)2 - b2
  • 8. 8 Everything Maths www.everythingmaths.co.za Factorisation ● Factorisation is the opposite process of expanding the brackets. ● Taking out a common factor is the basic factorisation method. ● We often need to use grouping in pairs to factorise polynomials. ● To factorise a quadratic we find the two binomials that were multiplied together to give the quadratic.
  • 9. 9 Everything Maths www.everythingmaths.co.za Difference and sum of two cubes ● The sum of two cubes can be factorised as: x3 + y3 = (x + y)(x2 − xy + y2 ) ● The difference of two cubes can be factorised as: x3 − y3 = (x − y)(x2 + xy + y2 )
  • 10. 10 Everything Maths www.everythingmaths.co.za Simplification of fractions ● We can simplify fractions by incorporating the methods we have learnt to factorise expressions. ● Only factors can be cancelled out in fractions, never terms. ● To add or subtract fractions, the denominators of all the fractions must be the same. a b × c d = ac bd , (b≠0,d≠0) a b + c b = a+c b , (b≠0) a b ÷ c d = a b × d c = ad bc , (b≠0,c≠0,d≠0)
  • 11. 11 Everything Maths www.everythingmaths.co.za For more practice see: www.everythingmaths.co.za