ALGEBRAIC EXPRESSIONS

ALGEBRAIC EXPRESSIONS
MODULE 1 ALGEBRAIC EXPRESSIONS

THE NUMBER SYSTEM REAL NUMBERS R RATIONALS Q IRRATIONALS INTEGERS Z
Grade 10 Module 1 Page 1 REAL NUMBERS R RATIONALS Q IRRATIONALS INTEGERS Z FRACTIONS WHOLE NUMBERS N "Textbook, Chapter, Page" NATURAL NUMBERS N

Grade 10 Module 1 Page 1 From early beginnings, the human race used counting numbers (natural numbers) in daily life. Eventually the number 0 was introduced and we had the whole numbers. The integers were introduced to cope with the concept of temperature (which can be negative) and overdrawn bank accounts. "Textbook, Chapter, Page"

was used for the fraction
Grade 10 Module 1 Page 1 In ancient times, simple fractions were introduced. For the Egyptians, the following symbols were used: was used for the fraction was used for the fraction "Textbook, Chapter, Page"

Grade 10 Module 1 Page 1 Eventually, the idea of a rational number was introduced. The word “rational” comes from the word “ratio”. For example, the statement “I have eaten of an apple” gives a comparison of the number of parts eaten to the total number of parts. 3 parts out of a total of 4 parts of the apple was eaten. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 1 The ratio is not an integer, but a new number between 0 and 1. It is called a rational number involving the division of the integers. "Textbook, Chapter, Page"

A rational number is a number that can be expressed in the form where
Grade 10 Module 1 Page 2 RATIONAL NUMBERS A rational number is a number that can be expressed in the form where and where a and b are integers. EXAMPLE 1 (a) All integers are rational numbers. "Textbook, Chapter, Page"

(b) Mixed fractions are rational numbers.
Grade 10 Module 1 Page 2 (b) Mixed fractions are rational numbers. (c) Terminating decimals are rational. "Textbook, Chapter, Page"

(d) Recurring decimals are rational numbers.
Grade 10 Module 1 Page 2 (d) Recurring decimals are rational numbers. A recurring decimal has an infinite pattern. For example: "Textbook, Chapter, Page"

(1) Show that is rational.
Grade 10 Module 1 Page 2 (1) Show that is rational. Now look for two equations where the decimals after the comma are the same and then subtract the equations. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 2 Two equations where the decimals are the same after the comma are (2) and (1). Subtract (1) from (2) as follows: "Textbook, Chapter, Page" rational

(2) Show that is rational.
Grade 10 Module 1 Page 3 (2) Show that is rational. Now look for two equations where the decimals after the comma are the same and then subtract the equations. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 3 Two equations where the decimals are the same after the comma are (3) and (1). Subtract (1) from (3) as follows: "Textbook, Chapter, Page" rational

The halfway mark between 1 and 2 is:
Grade 10 Module 1 Page 3 Rational numbers can be represented on a number line and there are infinitely many of them. To demonstrate this, let us keep on working out the halfway marks between 1 and 2. The halfway mark between 1 and 2 is: "Textbook, Chapter, Page"

The halfway mark between 1,5 and 2 is:
Grade 10 Module 1 Page 3 The halfway mark between 1,5 and 2 is: The halfway mark between 1,75 and 2 is: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 3 This process can be continued indefinitely and we can find an infinite number of rational numbers lying between 1 and 2 and that get closer to 2 all the time. Therefore, between any two rational numbers, there are an infinite number of rational numbers. "Textbook, Chapter, Page"

• Look at the digit to the right of this decimal place.
Grade 10 Module 1 Page 3 ROUNDING OFF NUMBERS The rules for rounding off numbers to certain decimal places are as follows: • Count to the number of decimal places after the comma that you want to round off to. • Look at the digit to the right of this decimal place. "Textbook, Chapter, Page"

If it is lower than 5, drop it and all the digits to the right of it.
Grade 10 Module 1 Page 3 If it is lower than 5, drop it and all the digits to the right of it. If it is 5 or more than 5, then add one digit to the digit immediately to the left of it and drop it and all the digits to the right of it. If necessary, keep or add zeros as place-holders. "Textbook, Chapter, Page"

Round off the following numbers to two decimal places:
Grade 10 Module 1 Page 4 EXAMPLE 2 Round off the following numbers to two decimal places: (a) 2,31437 2,31 (b) 0,7777 0,78 (c) 245,13589 245,14 (d) 245,2 245,20 "Textbook, Chapter, Page" (e) 11,4963 11,50

CALCULATOR OPERATIONS WITH RATIONAL NUMBERS
Grade 10 Module 1 Page 4 CALCULATOR OPERATIONS WITH RATIONAL NUMBERS EXAMPLE 3 Use a calculator to evaluate the following: (a) (3 decimal places) "Textbook, Chapter, Page"

(b) (2 decimal places) (c) (d) "Textbook, Chapter, Page"
Grade 10 Module 1 Page 4,5 (b) (2 decimal places) (c) (d) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 5 IRRATIONAL NUMBERS Pythagoras lived in Ancient Greece (570BC). He founded a secret mystical order called the Pythagoreans. Their motto was: Number rules the universe. The Pythagoreans had a high regard for rational numbers. The essence of their philosophy was that everything in nature should be expressible in terms of ratios of whole numbers. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 5 The Pythagoreans treated negative numbers as “absurd” quantities and avoided them altogether. It was only until the 7th century AD that negative numbers were mentioned and accepted as legitimate. The works of Brahamagupta, a Hindu mathematician, made extensive use of these numbers. "Textbook, Chapter, Page"

Only in the 17th century were negative numbers accepted universally.
Grade 10 Module 1 Page 6 Only in the 17th century were negative numbers accepted universally. The Pythagorean philosophy had its origin in music. Pythagoras had discovered that the common musical intervals produced by a vibrating string correspond to simple numerical ratios of string lengths. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 The Pythagoreans believed that the entire universe is constructed according to the laws of musical harmony, which involves fractions. Rational numbers dominated the Greek view of the world way back then. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 The Greeks realised that there are infinitely many integers as well as fractions. While integers have big gaps between them, gaps of one unit, the rational numbers are dense, meaning that between any two fractions, no matter how close they are, we can always find another fraction (rational number). "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 It seemed that the rational numbers form a massive, densely populated set of numbers, leaving no gaps between its members. This means that the entire number line is completely populated with rational numbers, or rational points. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 However, to their amazement, the Pythagoreans discovered that there are, in fact, “holes” or “gaps” between the rational numbers as well. They believed that this seemed to show something imperfect in the construction of the universe, as if the gods had overlooked something. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 Pythagoras encountered a problem when working with his famous theorem. Consider the following triangle: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 The number can easily be represented on a number line, as demonstrated by Pythagoras: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 6 Hence, Pythagoras concluded that must be a rational number, because every number on the number line is a rational number. However, to Pythagoras’ horror, he was not able to write as the ratio of two whole numbers. His conclusion was too terrible to even contemplate. He discovered that even though this number was on the number line, it was not a rational number. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 7 The world was eventually to learn of the existence of numbers referred to as irrational numbers. These numbers cannot be expressed in the form . Pythagoras discovered that these irrational numbers “fill the gaps” between the rational numbers on the number line. Collectively, the Rationals and Irrationals have been called the Real numbers. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 8 EXAMPLE 4 (a) Without using a calculator, determine between which two integers lies. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 8 (b) By calculating rounded off to four decimal places, determine between which two integers it lies. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 8 (c) Without using a calculator, determine between which two integers lies. "Textbook, Chapter, Page"

Represent the following sets on a number line:
Grade 10 Module 1 Page 9 SET BUILDER NOTATION EXAMPLE 5 Represent the following sets on a number line: (a) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 9 (b) (c) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 9 (d) (e) "Textbook, Chapter, Page"

Write the following in set builder notation:
Grade 10 Module 1 Page 9 EXAMPLE 6 Write the following in set builder notation: (a) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 10 (d) "Textbook, Chapter, Page"

Represent the following on a number line: (a)
Grade 10 Module 1 Page 10 INTERVAL NOTATION Interval notation is another way of representing real numbers on the number line. EXAMPLE 7 Represent the following on a number line: (a) "Textbook, Chapter, Page"

Write the following in interval notation:
Grade 10 Module 1 Page 10 (d) EXAMPLE 8 Write the following in interval notation: (a) "Textbook, Chapter, Page"

ALGEBRAIC EXPRESSIONS MULTIPLICATION OF ALGEBRAIC EXPRESSIONS
Grade 10 Module 1 Page 12 ALGEBRAIC EXPRESSIONS MULTIPLICATION OF ALGEBRAIC EXPRESSIONS THE DISTRIBUTIVE LAW (REVISION OF GR 9) To verify this law, we can substitute given numbers for the variables. Suppose that , and LHS "Textbook, Chapter, Page" RHS

THE PRODUCT OF TWO BINOMIALS Consider the product
Grade 10 Module 1 Page 12 THE PRODUCT OF TWO BINOMIALS Consider the product We can apply the distributive law as follows: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 12 This can also be done using the FOIL method. Here you must first multiply the first terms in each bracket. Then you multiply the outer terms, then the inner terms and finally the last terms. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 12 FIRSTS "Textbook, Chapter, Page"

Grade 10 Module 1 Page 12 OUTERS "Textbook, Chapter, Page"

Grade 10 Module 1 Page 12 INNERS "Textbook, Chapter, Page"

Grade 10 Module 1 Page 12 LASTS "Textbook, Chapter, Page"

Expand and simplify the following:
Grade 10 Module 1 Page 12 EXAMPLE 1 Expand and simplify the following: (a) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 12,13 (d) (e) "Textbook, Chapter, Page"

Consider the following expressions:
Grade 10 Module 1 Page 13 SQUARING A BINOMIAL Consider the following expressions: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 13 Therefore: "Textbook, Chapter, Page"

In other words, the pattern is as follows:
Grade 10 Module 1 Page 13 In other words, the pattern is as follows: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 13 EXAMPLE 1 Expand and simplify the following: (a) "Textbook, Chapter, Page"

DIFFERENCE OF TWO SQUARES Consider the following expression:
Grade 10 Module 1 Page 13 DIFFERENCE OF TWO SQUARES Consider the following expression: "Textbook, Chapter, Page"

In other words, the pattern is as follows:
Grade 10 Module 1 Page 13 In other words, the pattern is as follows: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 14 EXAMPLE 3 Expand and simplify the following: (a) (b) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 14 (c) (b) "Textbook, Chapter, Page"

THE PRODUCT OF A BINOMIAL AND A TRINOMIAL
Grade 10 Module 1 Page 15 THE PRODUCT OF A BINOMIAL AND A TRINOMIAL EXAMPLE 4 Expand and simplify the following: (a) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 15 (b) "Textbook, Chapter, Page"

FACTORISATION OF ALGEBRAIC EXPRESSIONS
Grade 10 Module 1 Page 15 FACTORISATION OF ALGEBRAIC EXPRESSIONS TAKING OUT THE HIGHEST COMMON FACTOR Factorisation is the reverse of multiplication. Consider the reverse procedure of the distributive law: "Textbook, Chapter, Page"

Factorise the following expressions: (a) The factors of 12 are:
Grade 10 Module 1 Page 15 EXAMPLE 5 Factorise the following expressions: (a) The factors of 12 are: The factors of 8 are: Therefore the highest common factor between 12 and 8 is 4. "Textbook, Chapter, Page"

The expression can be written as
Grade 10 Module 1 Page 15,16 The expression can be written as The expression can be written as Therefore the highest common factor between and is Therefore the highest common factor between and is "Textbook, Chapter, Page"

Grade 10 Module 1 Page 16 We can now take out the highest common factor (HCF) as follows and thus factorise the expression by writing it as a product of terms: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 16 (c) "Textbook, Chapter, Page"

Before discussing the next example, consider the expressions:
Grade 10 Module 1 Page 16 The next example involves changing the sign before a bracket before taking out the common bracket. Before discussing the next example, consider the expressions: "Textbook, Chapter, Page"

Therefore it is true that:
Grade 10 Module 1 Page 16 Therefore it is true that: You can write this as: For example: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 16 Therefore, if you change the sign before the bracket, change the signs of the terms when writing them in the brackets. "Textbook, Chapter, Page"

EXAMPLE 6 Factorise: (a) "Textbook, Chapter, Page"
Grade 10 Module 1 Page 16 EXAMPLE 6 Factorise: (a) "Textbook, Chapter, Page"

DIFFERENCE OF TWO SQUARES
Grade 10 Module 1 Page 17 DIFFERENCE OF TWO SQUARES Consider the product The reverse process is called the factorisation of the difference of two squares. "Textbook, Chapter, Page"

Another way of seeing this type of factorisation is:
Grade 10 Module 1 Page 17 Another way of seeing this type of factorisation is: "Textbook, Chapter, Page"

EXAMPLE 7 Factorise fully: (a) "Textbook, Chapter, Page"
Grade 10 Module 1 Page 17 EXAMPLE 7 Factorise fully: (a) "Textbook, Chapter, Page"

By multiplying out, it is clear that this product will become:
Grade 10 Module 1 Page 18 QUADRATIC TRINOMIALS Consider the product By multiplying out, it is clear that this product will become: "Textbook, Chapter, Page"

For example, the trinomial can be factorised as follows:
Grade 10 Module 1 Page 18 So the expression can be factorised as For example, the trinomial can be factorised as follows: Write the last term, 8, as the product of two numbers The options are: "Textbook, Chapter, Page"

because the sum of the numbers 4 and 2 gives 6. Therefore:
Grade 10 Module 1 Page 18 The middle term is now obtained by adding the numbers of one of the above options. The obvious choice will be the option because the sum of the numbers 4 and 2 gives 6. Therefore: "Textbook, Chapter, Page"

So the trick to factorising trinomials is as follows:
Grade 10 Module 1 Page 18 So the trick to factorising trinomials is as follows: • Write down the last term as the product of two numbers. • Find the two numbers (using the appropriate numbers from one of the products) which gets the middle term by adding or subtracting. "Textbook, Chapter, Page"

• Check that when you multiply these numbers you get the last term.
Grade 10 Module 1 Page 18 • Check that when you multiply these numbers you get the last term. "Textbook, Chapter, Page"

The last term can be written as the following products:
Grade 10 Module 1 Page 18 EXAMPLE 8 Factorise: (a) The last term can be written as the following products: "Textbook, Chapter, Page"

We now need to get from one of the options above.
Grade 10 Module 1 Page 18 We now need to get from one of the options above. Using will enable us to get since (middle term) and (last term) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 19 Therefore: "Textbook, Chapter, Page"

The last term can be written as the following products:
Grade 10 Module 1 Page 19 (b) The last term can be written as the following products: We now need to get from one of the options above. Try the option "Textbook, Chapter, Page"

Clearly which is the middle term and which is the last term.
Grade 10 Module 1 Page 19 Clearly which is the middle term and which is the last term. Notice that the option will not work because even though is the middle term, is not the last term. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 19 Therefore: Notice: • If the sign of the last term of a trinomial is negative, the signs in the brackets are different (see example 8a). "Textbook, Chapter, Page"

Grade 10 Module 1 Page 19 • If the sign of the last term of a trinomial is positive, the signs in the brackets are the same i.e. both positive or both negative (see example 8b). "Textbook, Chapter, Page"

Here it is necessary to first take out the highest common factor:
Grade 10 Module 1 Page 19 (c) Here it is necessary to first take out the highest common factor: "Textbook, Chapter, Page"

• Take out the highest common factor if necessary.
Grade 10 Module 1 Page 20 In summary then, apply the following procedure when factorising trinomials: • Take out the highest common factor if necessary. • Write down the last term as the product of two numbers. "Textbook, Chapter, Page"

• Check that when you multiply these numbers you get the last term.
Grade 10 Module 1 Page 20 • Find the two numbers (using the appropriate numbers from one of the products) which gets the middle term by adding or subtracting. • Check that when you multiply these numbers you get the last term. "Textbook, Chapter, Page"

Grade 10 Module 1 Page 20 • If the sign of the last term of a trinomial is positive, the signs in the brackets are the same (both positive or both negative). • If the sign of the last term of a trinomial is negative, the signs in the brackets are different. "Textbook, Chapter, Page"

MORE ADVANCED TRINOMIALS EXAMPLE 9 Consider the trinomial
Grade 10 Module 1 Page 21 MORE ADVANCED TRINOMIALS EXAMPLE 9 Consider the trinomial The method to factorise this trinomial is a little more involved than with the previous trinomials. A suggested method is as follows. "Textbook, Chapter, Page"

Write down the product options of the first and last terms
Grade 10 Module 1 Page 21 Step 1 Write down the product options of the first and last terms "Textbook, Chapter, Page"

Write the product options in a table format as follows:
Grade 10 Module 1 Page 21 Step 2 Write the product options in a table format as follows: First term Last term The product option for the last term must also be written in reverse order as "Textbook, Chapter, Page"

Grade 10 Module 1 Page 21 Step 3 Select any product option from the first term and last term and write these options using what is called the “cross method”: "Textbook, Chapter, Page"

Now multiply as follows:
Grade 10 Module 1 Page 21 Step 4 Now multiply as follows: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 22 Step 5 The trick is to now get the middle term, , from and using different signs (because the sign of the last term of the trinomial is negative). Insert the signs as follows: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 22 "Textbook, Chapter, Page"

The factors are now obtained by reading off horizontally:
Grade 10 Module 1 Page 22 Step 6 The factors are now obtained by reading off horizontally: "Textbook, Chapter, Page"

The product options of the first and last terms:
Grade 10 Module 1 Page 22 EXAMPLE 10 Factorise: The product options of the first and last terms: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 21 First term Last term The signs in the brackets must be the same because the sign of the last term of the trinomial is positive. "Textbook, Chapter, Page"

The product options of the first and last terms:
Grade 10 Module 1 Page 23 EXAMPLE 11 Factorise: The product options of the first and last terms: "Textbook, Chapter, Page"

First term Last term "Textbook, Chapter, Page"
Grade 10 Module 1 Page 23 First term Last term "Textbook, Chapter, Page"

FACTORISATION BY GROUPING IN PAIRS EXAMPLE 12 Factorise: Method 1
Grade 10 Module 1 Page 24 FACTORISATION BY GROUPING IN PAIRS EXAMPLE 12 Factorise: Method 1 Group the first two terms together and the last two together. Factorise each pair separately and then take out the common bracket: "Textbook, Chapter, Page"

Grade 10 Module 1 Page 24 Method 2 Group the first and third terms together and the second and fourth terms together. Factorise each pair separately and then take out the common bracket: "Textbook, Chapter, Page"

It is clear from the above that:
Grade 10 Module 1 Page 24 Before discussing the next examples, we need to deal with the concept of “taking out a negative”. Consider the following expressions: It is clear from the above that: "Textbook, Chapter, Page"

middle sign is positive and changes to a negative in the brackets
Grade 10 Module 1 Page 24 Therefore, whenever you “take out a negative sign” when factorising an expression, the middle sign will always change in the brackets. For example: middle sign is positive and changes to a negative in the brackets "Textbook, Chapter, Page"

middle sign is negative and changes to a positive in the brackets
Grade 10 Module 1 Page 24 middle sign is negative and changes to a positive in the brackets middle sign is negative and changes to a positive in the brackets "Textbook, Chapter, Page"

Factorise the following expressions fully:
Grade 10 Module 1 Page 25 EXAMPLE 13 Factorise the following expressions fully: (a) "Textbook, Chapter, Page"

SUM AND DIFFERENCE OF TWO CUBES Consider the following products: (a)
Grade 10 Module 1 Page 26 SUM AND DIFFERENCE OF TWO CUBES Consider the following products: (a) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 26 (f) (g) "Textbook, Chapter, Page"

Grade 10 Module 1 Page 26 Conclusion: "Textbook, Chapter, Page"

Factorise the following:
Grade 10 Module 1 Page 26 EXAMPLE 14 Factorise the following: (a) "Textbook, Chapter, Page"

SIMPLIFICATION OF ALGEBRAIC FRACTIONS
Grade 10 Module 1 Page 28 SIMPLIFICATION OF ALGEBRAIC FRACTIONS MULTIPLICATION AND DIVISION EXAMPLE 15 Simplify the following expressions: (a) "Textbook, Chapter, Page"

Simplify the following expressions:
Grade 10 Module 1 Page 29 EXAMPLE 16 Simplify the following expressions: (a) "Textbook, Chapter, Page"

EXAMPLE 17 Simplify: (a) "Textbook, Chapter, Page"
Grade 10 Module 1 Page 30 EXAMPLE 17 Simplify: (a) "Textbook, Chapter, Page"

EXAMPLE 18 Simplify: "Textbook, Chapter, Page"
Grade 10 Module 1 Page 31 EXAMPLE 18 Simplify: "Textbook, Chapter, Page"

EXAMPLE 19 Simplify: "Textbook, Chapter, Page"
Grade 10 Module 1 Page 32 EXAMPLE 19 Simplify: "Textbook, Chapter, Page"

EXAMPLE 20 Simplify: Method 1 "Textbook, Chapter, Page"
Grade 10 Module 1 Page 32 EXAMPLE 20 Simplify: Method 1 "Textbook, Chapter, Page"

Grade 10 Module 1 Page 32 Method 2 "Textbook, Chapter, Page"

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