Everything Maths Grade 10. Chapter 1: Algebraic expressions.

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1. Algebraic expressions
Grade 10

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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; . . . }

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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.

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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

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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

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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.

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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

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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.

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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
)

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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)

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