An algebraic expression is a description of a set of operations that are to be done in a certain order. In this chapter, you will learn to specify a different set of operations that will produce the same results as a given set of operations. Two different expressions that produce the same results are called equivalent expressions.
Algebraic language
Words, diagrams and expressions
 Complete this
table.
Words
Flow diagram
Expression
Multiply a number by 5 and then subtract 3 from the answer.
\(5x  3\)
(a)
Add 5 to a number and then multiply the answer by 3.
(b)
(c)
\(3(2x + 3)\)
An algebraic expression indicates a sequence of operations that can also be described in words. In some cases they can be described with flow diagrams.
Expressions in brackets should always be calculated first. If there are no brackets in an algebraic expression, it means that multiplication and division must be done first, and addition and subtraction afterwards.
For example, if \(x = 5\) the expression \(12 + 3x\) means "multiply 5 by 3, then add 12". It does not mean "add 12 and 3, then multiply by 5".
If you wish to say "add 12 and 3, then multiply by 5", the numerical expression should be \(5 \times (12 + 3)\) or \((12 + 3) \times 5\).
 Describe each of these
sequences of calculations with an algebraic expression:
 Multiply a number by 10, subtract 5 from
the answer, and multiply the answer by 3.
 Subtract 5 from a number, multiply the
answer by 10, and multiply this answer by 3.
 Multiply a number by 10, subtract 5 from
the answer, and multiply the answer by 3.
 Evaluate each of these
expressions for x = 10:
 \(200  5x\)
 \((200  5)x\)
 \(5x + 40\)
 \(5(x + 40)\)
 \(40 + 5x\)
 \(5x + 5 \times 40\)
 \(200  5x\)
Some words we use in algebra
An expression with one term only, like \(3x^2\), is a monomial.
An expression which is a sum of two terms, like \(5x + 4\), is called a binomial.
An expression which is a sum of three terms, like \(3x^3 + 2x + 9\), is called a trinomial.
The symbol x is often used to represent the variable in an algebraic expression, but other letter symbols may also be used.
In the monomial \(3x^2\), the 3 is called the coefficient of \(x^2\).
In the binomial \(5x + 4\), and the trinomial \(3x^2 + 2x + 9\) the numbers 4 and 9 are called constants.
 Complete the table, using the
completed first row as an example.
Expression
Type of expression
Symbol used to represent the variable
Constant
Coefficient of
\(x^2 + 6x + 10\)
Trinomial
\(x\)
10
the second term is 6
\(6s^3 + s^2 + 5\) \(s^2\) is ______
\(\frac{k}{3} + 12\) the first term is
\(4p^{10}\)
\(p^{10}\) is _______
 Consider the
patternpolynomial starting with \(7x^5 + 5x^4 + 3x^3 + x^2 +\) ...
 What is the coefficient of
the fourth term?
 What is the exponent value
of the fifth term?
 Do you think the sixth term will be a
constant? Why?
 What is the coefficient of
the fourth term?
Equivalent algebraic expressions
 Calculate the
numerical value of the expressions for the various values of
\(x\). Do the
calculations in your exercise book. Then fill in your
answers.
\(x\)
2
1
0
1
2
(a)
\(3x + 2\)
(b)
\(2x  3\)
(c)
\(3x + 2 + 2x  3\)
(d)
\(2x  3 + 3x + 2\)
(e)
\(5x  1\)
(f)
\((3x + 2)(2x  3)\)
(g)
\(3x(2x  3) + 2(2x  3)\)
(h)
\(6x^2  5x  6\)
(i)
\(\frac{(3x + 2)(2x3)}{3x + 2}\)
(j)
\(\frac{6x^2 + 5x  6}{3x + 2}\)
 Make a list of all the
algebraic expressions above which have the same numerical value
for the same value of \(x\), although they may look
different:
Equivalent expressions are algebraic expressions that have different sequences of operations, but have the same numerical value for any given value of \(x\).
It is often convenient not to work with a given expression, but to replace it with an equivalent expression.
 Complete this table.
\(x\)
2
3
5
10
5
10
\(12x  7 + 3x + 10  5x\)
 Complete this table.
\(x\)
2
3
5
10
5
10
\(10x + 3\)

 Is \(10x + 3\) equivalent to
\(12x  7 + 3x + 10  5x\)?
Explain your answer.

Suppose you need to know how much \(12x  7 + 3x +
10  5x\) is for \(x = 37\) and \(x = 43\). What do you think is the
easiest way to find out?
 Is \(10x + 3\) equivalent to
\(12x  7 + 3x + 10  5x\)?
Explain your answer.
Conventions for writing algebraic expressions
Here are some things that mathematicians have agreed upon, and it makes mathematical work much easier if all people stick to these agreements.
A convention is something that people have agreed to do in the same way.
The multiplication sign is often omitted in algebraic expressions: We normally write \(4x\) instead of \(4 \times x\) and \(4(x  5)\) instead of \(4 \times (x  5)\). It is a convention to write a known number first in a product, i.e. we write 3 Ã— x rather than x Ã— 3, and we write 3x but not x3.
 Rewrite each of the
following in the way in which it is normally written in
algebraic expressions.
 \(x \times 4 + x \times
y  y \times 3\)
 \(7 \times
(10  x) + (5 \times x +
3)10\)
 \(x \times 4 + x \times
y  y \times 3\)
People all over the world have agreed that, in expressions that do not contain brackets, addition and subtraction should be performed as they appear from left to right in the expression.
According to this convention, \(x  y + z\) means that you first have to subtract \(y\) from \(x\), then add \(z\). For example if \(x = 10\), \(y = 5\) and \(z = 3\), \(x  y + z\) is \(10  5 + 3\) and it means \(10  5 = 5\), then \(5 + 3 = 8\). It does not mean \(5 + 3 = 8\), then \(10  8 = 2\).
 Calculate \(50  20 + 30\) and \(50
+ 30  20\) and \(50  30 + 20\)
 Evaluate each of the following
expressions for \(x = 10, ~y = 5\) and \(z =2 \).
 \(x + y z\)
 \(x z +y\)
 \(10y  3x + 5z  4y \)
 \(10y  3x  5z + 4y + 3x\)
 \(x + y z\)
People have also agreed that, in expressions that do not contain brackets, we should do multiplication (and division) before addition and subtraction.
Hence \(5 + 3 \times 4\) should be understood as "multiply 4 by 3, then add the answer to 5" and not as "add 5 and 3 then multiply the answer by 4".
Also, \(3 \times 4 + 5\) should be understood to mean "multiply 4 by 3, then add 5 to the answer", and not as "add 4 and 5 then multiply the answer by 3".
 Do each of the following
calculations.
 multiply 4 by 3, then add 5 to the
answer
 add 4 and 5 then multiply the answer by
3
 multiply 4 by 3, then add the answer to
5
 add 5 and 3 then multiply the answer by
4
 multiply 4 by 3, then add 5 to the
answer
 Rewrite the instructions in
4(a) and 4(c) without using words.
 Calculate each of the
following.
 \(10 \times 5 + 30\)
 \(30 + 10 \times 5\)
 \(10 \times 5  30\)
 \(30  10 \times 5\)
 \(10 \times 5 + 30\)

 Add 4 and 5, then subtract the answer from 20.
 Subtract 4 from 20 and then add 5.
 Add 4 and 5, then multiply the answer by 3.
 Multiply 3 by 5 and then add the answer to 4.
 Add 4 and 5, then subtract the answer from 20.
If we want to specify the calculations in 7(a) and 7(c) without using words we face challenges.
We cannot write 20  4 + 5 for "add 4 and 5 then subtract the answer from 20", because that would mean "subtract 4 from 20 then add 5". We need a way to indicate, without using words, that we want the addition to be performed before the subtraction in this case.
Similarly we cannot write 4 + 5 \(\times\) 3 for "add 4 and 5 then multiply the answer by 3", because that would mean "multiply 3 by 5 and then add the answer to 4". We need a way to indicate, without using words, that we want the addition to be performed before the multiplication in this case.
Mathematicians have agreed to use brackets to address the above challenges. The following convention is used all over the world:
Whenever there are brackets in an expression, the calculations within the brackets should be performed first.
Hence \(20  (4 + 5)\) means add 4 and 5 then subtract the answer from 20, but 20  4 + 5 means subtract 4 from 20 then add 5.
\((4 + 5) \times 3\) or \(3 \times (4 + 5)\) means add 4 and 5 then multiply the answer by 3, but \( 4 + 5 \times 3\) means multiply 3 by 5 then add the answer to 4.
\(10 + 2(5 + 9)\) means add 5 and 9, multiply the answer by 2, then add this answer to 10:
\(5 + 9 = 14 \\ 14 \times 2 = 28 \\ 28 + 10 = 38 \)
 Calculate each of the
following.
 \(100 + 50  30\)
 \(100 + (50  30)\)
 \(100  50 + 30 \)
 \(100  (50 + 30)\)
 \(3(10  4) + 2 \)
 \(10(5 + 7) + 3(18  8)\)
 \(250  10 \times (18 + 2) + 35 \)
 \((20 + 20) \times (20  10)\)
 \((250  10) \times (18 + 2) + 35 \)
 \(20 + 20 \times (20  10)\)
 \(200 + (100 \times 2(15 + 5)) \)
 \((200 + 100) \times 2 \times 15 + 5\)
 \(100 + 50  30\)
In algebra, we normally write \(3(x + 2y)\) instead of \((x + 2y) \times 3\), and we write \(3(x  2y)\) instead of \((x  2y) \times 3\). Don't let this conventional way of writing in algebra confuse you. The expression \(3(x + 2y)\) does not mean that multiplication by 3 is the first thing you should do when you evaluate the expression for certain values of x and y. The first thing you should do is to add the values of x and y. That is what the brackets tell you!
However, performing the instructions 3(x + 2y) is not the only way in which you can find out how much 3(x + 2y) is for any given values of x and y. Instead of working out 3(x + 2y), you may work out 3x + 6y. In this case you will multiply each term before you add them together.
 Evaluate each of the following
expressions for \(x = 10,~y = 5\) and \(z = 2\).
 \(xy + z\)
 \(x(y + z)\)
 \(x + yz \)
 \(xy + xz\)
 \(xy  z\)
 \(x(y  z)\)
 \( x  yz\)
 \(xy  yz\)
 \(x + (y  z) \)
 \(x  (y  z)\)
 \(x  (y + z)\)
 \(x  y  z\)
 \(x + y  z\)
 \(x  y + z\)
 \(xy + z\)
Properties of operations
 Calculate the following:
 \(5(3 + 4) \)
 \(5 \times 3 + 5 \times 4\)
 \(6 \times 3 + (4 + 6)\)
 \(6 + 4) + 3 \times 6\)
 \(3 \times (4 \times 5) \)
 \((3 \times 4) \times 5\)
 \(5(3 + 4) \)
You should have noticed that for each row the results are the same. This is because operations with numbers have certain properties, namely the distributive, commutative and associative properties.
The distributive property is used each time you multiply a number in parts. For example:
The number thirtyfour is actually \(30 + 4\). You may calculate \(5 \times 34\) by calculating \(5 \times 30\) and \(5 \times 4\), and then adding the two answers:
\(5 \times 34 = 5 \times 30 + 5 \times 4\)
The word "distribute" means to spread out. The distributive property may be described as follows:
\(a(b + c) = ab + ac\)
where \(a,~b\) and \(c\) can be any numbers.
We may say: "multiplication distributes over addition"
 Calculate each of the
following:
 \(5(x  y)\) for \(x = 10\) and \(y = 8 \)
 \(5x  5y\) for \(x = 10\) and \(y = 8\)
 \(5(x  y)\) for \(x = 100\) and \(y = 30 \)
 \(5x  5y\) for \(x = 100\) and \(y = 30\)
 \( 5(x  y + z)\) for \(x = 10, ~y = 3,~z = 2 \)
 \(5x  5y + 5z\) for \(x = 10, ~y = 3, ~z = 2\)
 \(5(x  y)\) for \(x = 10\) and \(y = 8 \)
 We say "multiplication
distributes over addition".
Does multiplication also distribute over subtraction?
Give examples to support your answer.
For any values of \(x\) and \(y\),
 \(x + y\) and \(y + x\) give the same answers, and
 \( xy\) and \(yx\) give the same answers.
This is called the commutative property of addition, and multiplication.
 We say "addition is
commutative" and "multiplication is commutative".
Is subtraction also commutative? Demonstrate your answer with an example.
The associative property allows you to arrange three or more numbers in any sequence when adding or multiplying. For any values of x, y and z, the following expressions all have the same answer:
\(x + y + z \\ y + x + z \\ z + y + x\)
 Calculate 16 + 33 + 14 + 17 in
the easiest possible way.
The associative property of multiplication allows you to simplify something like the following.
\(abc + bca + cba\)
Because the order of multiplication does not change the result we can rewrite this expression as: \(abc + abc + abc\).
This then can be simplified by adding like terms to be \(3abc\). You will be able to use these properties throughout this chapter and when you do algebraic manipulations.
When you form an expression that is equivalent to a given expression you say that you manipulate the expression.
 Replace each of the
following expressions with a simpler expression that will give
the same answer. Do not do any calculations now. In each
case state why your replacement will be easier to do.
 \( 17 \times 43 + 17 \times 57\)
 \( 7 \times 5 \times 8 \times 4 + 12 \times
8 \times 4 \times 7  9 \times 4 \times 5 \times 8\)
 \( 43 \times 17 + 57 \times 17\)
 \(43x + 57x\) (for \(x = 213\) or any other value)
 \( 17 \times 43 + 17 \times 57\)
 Which properties of operations
did you use in each part of question 6?
Combining like terms in algebraic expressions
Rearrange terms, then combine like terms
To check whether two expressions are possibly equivalent, you can evaluate both expressions for several different values of the variable.
 In each case below,
first predict whether the two expressions are equivalent and
then check by evaluating both for \(x = 1,~ x = 10,~ x = 2\) and \(x = 2\) in the tables.
 \(x(x + 3)\) and \(x^2 + 3\)
 \(x(x + 3)\) and \(x^2 + 3x\)
 \(x(x + 3)\) and \(x^2 + 3\)
Some expressions can be simplified by rearranging the terms and combining "like terms".
In the expression \(5x^2 + 13x + 7 + 2x^2  8x  12\), the terms \(5x^2\) and \(2x\) are like terms.
Two or more like terms can be combined to form a single term.
\(5x^2 + 2x^2\) can be replaced by \(7x^2\) because for any value of \(x\), for example \(x = 2\) or \(x = 10\), calculating \(5x^2 + 2x^2\) and \(7x^2\) will produce the same output value (try it!).
 Complete the table.
\(x\)
10
2
5
1
\(5x^2 + 2x^2\)
\(7x^2\)
\(13x  8x\)
\(5x\)
It is difficult to see the like terms in a long expression like 3x2 + 13x + 7 + 2x2  8x  12. Fortunately, you can rearrange the terms in an expression so that the like terms are next to each other.

 Complete the second and third rows of
the table below. You will complete the next two rows when you
do question (g).
\(x\)
10
2
5
1
\(3x^2 + 13x + 7 + 2x^2  8x  12\)
\(3x^2 + 2x^2 + 13x  8x + 7  12\)
 What do you observe?
 How does the one expression in the above
table differ from the other one?
 Combine like terms in \(3x^2 + 2x^2 + 13x  8x + 7  12\) to make a shorter equivalent
expression.
 Evaluate your shorter expression for
\(x = 10,~ x = 2\) and \(x = 5\).
 Is your shorter expression equivalent to
\(3x^2 + 13x + 7 + 2x^2  8x  12\)?
Explain how you know whether it is or is not.
 Evaluate \(5x^2 + 5x  5\) and \(5(x^2 + x  1)\) for \(x = 10,~ x = 2, x = 5\) and \(x = 1\), and write your answers in the last two rows of the above table.
 Complete the second and third rows of
the table below. You will complete the next two rows when you
do question (g).
 Simplify each expression:
 \((3x^2 + 5x + 8) + (5x^2 + x + 4)\)
 \((7x^2 + 3x + 5) + (2x^2  x  2)\)
 \(6x^2  7x  4) + (4x^2 + 5x + 5)\)
 \((2x^2  5x  9)  (5x^2  2x  1)\)
 \((2x^2 + 5x  3) + (3x^2  9x + 5)\)
 \((y^2 + y + 1) + (y^2  y  1)\)
 \((3x^2 + 5x + 8) + (5x^2 + x + 4)\)
 Complete the table. (Hint:
Save yourself some work by simplifying first!)
\(x\)
2,5
3,7
6,4
12,9
35
4,7
0,04
\((3x + 6,5) + (7x + 3,5)\)
\((13x  6) + (26  12x)\)
 Simplify:
 \((2r^2 + 3r  5) + (7r^2  8r  12)\)
 \((2r^2 + 3r  5)  (7r^2  8r  12)\)
 \((2x + 5xy + 3y)  (12x  2xy  5y)\)
 \((2x + 5xy + 3y) + (12x  2xy  5y)\)
 \((2r^2 + 3r  5) + (7r^2  8r  12)\)
 Evaluate the following
expressions for\( x = 3,~ x = 2, ~x = 5\) and
\(x = 3\).
 \( 2x(x^2  x  1) + 5x(2x^2 + 3x  5)  3x(x^2 + 2x + 1)\)
 \( (3x^2  5x + 7)  (7x^2 + 3x  5) + (5x^2  2x + 8)\)
 \( 2x(x^2  x  1) + 5x(2x^2 + 3x  5)  3x(x^2 + 2x + 1)\)
 Write equivalent expressions
without brackets.
 \( 3x^4  (x^2 + 2x)\)
 \(3x^4  (x2  2x)\)
 \( 3x^4 + (x^2 + 2x)\)
 \( x  (y + z  t))\)
 \( 3x^4  (x^2 + 2x)\)
 Write equivalent
expressions without brackets, rearrange so that like terms are
grouped together, and then combine the like terms.
 \( 2y^2 + (y2  3y)\)
 \( 3x^2 + (5x + x^2)\)
 \(6x^2  (x4 + 3x^2)\)
 \(2t^2  (3t^2  5t^3)\)
 \( 6x^2 + 3x  (4x^2 + 5x)\)
 \(2r^2  5r + 7 + (3r^2  7r  8)\)
 \( 5(x^2 + x) + 2(x^2 + 3x)\)
 \( 2x(x  3) + 5x(x + 2)\)
 \( 2y^2 + (y2  3y)\)
 Write equivalent expressions
without brackets and simplify these expressions as far as
possible.
Example
\( \begin{align} 5r^2  2r(r + 5) &= 5r^2  2r^2  10r \\ &= 3r^2  10r \end{align}\)
 \(3x^2 + x(x + 3) \)
 \( 5x + x(7  2x) \)
 \( 6r2  2r(r  5) \)
 \( a(a + 3) + 5a(a  2) \)
 \( 6y(y + 1)  3y(y + 2) \)
 \( 4x(2x  3)  3x(x + 2) \)
 \( 2x^2(x  5)  x(3x^2  2) \)
 \(x(x  1) + x(2x + 3)  2x(3x + 1) \)
 \(3x^2 + x(x + 3) \)
Multiplication of algebraic expressions
Multiply polynomials by monomials

 Calculate \(3 \times 38\) and \(3 \times 62\)
and add the two answers.
 Add 38 and 62, then multiply the answer
by 3.
 If you do not get the same answer for (a) and (b), you have made a mistake. Rework until you get it right.
 Calculate \(3 \times 38\) and \(3 \times 62\)
and add the two answers.
The fact that if you work correctly, you get the same answer in questions 1(a) and 1(b), is a demonstration of the distributive property.
The distributive property may be described as follows:
\(a(b + c) = ab + ac\) and
\(a(b  c) = ab  ac\),
where \(a, ~b\) and \(c\) can be any numbers.
What you saw in question 1 was that \(3 \times 100 = 3 \times 38 + 3 \times 62\).
This can also be expressed by writing \(3(38 + 62) = 3 \times 38 + 3 \times 62\).

 Calculate \(10 \times 56\)
 Calculate \(10 \times 16 + 10 \times 40\)
 Calculate \(10 \times 56\)

 Write down any two numbers smaller
than 100. Let us call them x andy. Add your two
numbers, and multiply the answer by 3.
 Calculate \(3 \times x \) and \(3 \times
y\) and add the two answers.
 If you do not get the same answers for (a) and (b) you have made a mistake somewhere. Correct your work.
 Write down any two numbers smaller
than 100. Let us call them x andy. Add your two
numbers, and multiply the answer by 3.
 Complete
the table.
\(x\)
12
50
5
\(y\)
4
30
10
\(5x  5y\)
\(5(x  y)\)
\(5x + 5y\)
\(5(x + y)\)
Performing the instructions \(5(x + y)\) is not the only way in which you can find out how much \(5(x + y)\) is for any given values of \(x\) and \(y\). Instead of doing \(5(x + y)\) you may do \(5x + 5y\). In this case you will multiply first, and again, before you add.

 For \(x = 10\) and \(y = 20\),
evaluate \(8(x +y)\) by first adding 10 and 20, and
then multiplying by 8.
 Now evaluate \(8(x +y)\) by
doing \(8x + 8y\), in other words first calculate \(8
\times 10\) and \(8 \times 20\).
 For \(x = 10\) and \(y = 20\),
evaluate \(8(x +y)\) by first adding 10 and 20, and
then multiplying by 8.
 In question 5 you
evaluated \(8(x +y)\) in two different
ways for the given values of \(x\) and \(y\). Now also
evaluate \(20(x  y)\) in two different ways, for
\(x = 5\) and \(y = 3\).
 Use the distributive
property in each of the following cases to make a different
expression that is equivalent to the given expression.
 \(a(b + c) \)
 \(a(b + c + d) \)
 \(x(x + 1) \)
 \(x(x^2 + x + 1)\)
 \( x(x^3 + x^2 + x + 1) \)
What you do in this question is sometimes called "multiplication of a polynomial by a monomial".
One may also say that in each case you expand the expression, or you write an equivalent expression in expanded form.
 \(x^2(x^2  x + 3) \)
 \( 2x^2(3x2 + 2)\)
 \(3x^3(2x^2 + 4x  5) \)
 \( 2x^4(x^3  2x^2  4x + 5) \)
 \(a^2b(a^3  a^2 + a + 1) \)
 \(x^2y^3(3x^2y + xy^2  y) \)
 \( 2x(x^3  y^3) \)
 \( 2a^2b(3a^2 + 2a^2b^2 + 4b^2) \)
 \(2ab^2(3a^3  1) \)
 \(a(b + c) \)
 Expand the parts of
each expression and simplify.
Then evaluate the expression for \(x = 5\).
 \( 5(x  2) + 3(x + 4) \)
 \( x(x + 4)  4 (x + 4)\)
 \(x(x  4) + 4(x  4) \)
 \( x(x^2 + 3x + 9)  3(x^2 + 3x + 9) \)
 \( x(x^2  3x + 9) + 3(x^2  3x + 9) \)
 \(x^2(x^2  3x + 4)  x(x^3 + 4x2 + 2x + 3) \)
 \( 5(x  2) + 3(x + 4) \)
 Write in expanded form.
 \( x(x^2 + 2xy + y^2) + y(x^2 + 2xy + y^2) \)
 \( x^2y(x^2  2xy + y^2)  xy^2(2x^2  3xy  y^2) \)
 \( ab^2c(b^2c^2  ac) + b^2c^4(a^2 + abc^2) \)
 \(p^2q(pq^2 + p + q) + pq(p  q^2) \)
 \( x(x^2 + 2xy + y^2) + y(x^2 + 2xy + y^2) \)
Squares and cubes and roots of monomials
 Evaluate each of the following
expressions for \(x = 2, ~x = 5\) and \(x = 10\).
 \( (3x)^2 \)
 \(9x^2\)
 \((2x)^2\)
 \(4x^2 \)
 \((2x)^3\)
 \(8x^3\)
 \( (2x + 3x)^2 \)
 \((10x  7x)^2 \)
 \( (3x)^2 \)
 In each case, write an
equivalent monomial without brackets.
 \((5x)^2 \)
 \((5x)^3\)
 \((20x)^2\)
 \( (10x)^3 \)
 \( (2x + 7x)^2 \)
 \( (20x  13x)^3\)
 \((5x)^2 \)
The square root of \(16x^2\) is \(4x\), because \((4x)^2 = 16x^2\).
 Write down the square root of
each of the following expressions.
 \(\sqrt{(7x)^2}\)
 \(\sqrt{(9x)^2}\)
 \(\sqrt{(20x)^2}\)
 \(\sqrt{100x^2}\)
 \(\sqrt{(20x 15x)^2}\)
 \(\sqrt{25x^2}\)
 \(\sqrt{(21x  16x )^2}\)
 \(\sqrt{(5x)^2}\)
 \(\sqrt{(7x)^2}\)
The cube root of \(64x^3\) is \(4x\), because \((4x)^3 = 64x^3\)
 Write down the square root of
each of the following expressions.
 \(\sqrt[3]{(7x)^3}\)
 \(\sqrt[3]{(27x)^3}\)
 \(\sqrt[3]{(20x)^3}\)
 \(\sqrt[3]{1000x^3}\)
 \(\sqrt[3]{(20x  15x)^3}\)
 \(\sqrt[3]{125x^3}\)
 \(\sqrt[3]{(7x)^3}\)
Dividing polynomials by integers and monomials
 Complete
the table.
\(x\)
20
10
5
5
10
20
\((100x  5x^2) \div 5x\)
\(20  x\)
Can you explain your observations?

 R240 prize money must be shared
equally between 20 netball players. How much should each one
get?
 Mpho decided to do the calculations
below. Do not do Mpho's calculations, but think about this:
Will Mpho get the same answer that you got for question
(a)?
\((140 \div 20) + (100 \div 20)\)
 (c) Gert decided to do the calculations
below. Without doing the calculations, say whether Gert will
get the same answer that you got for question (a).
\((240 \div 12) + (240 \div 8)\)
 R240 prize money must be shared
equally between 20 netball players. How much should each one
get?
 Do the necessary
calculations to find out whether the following statement are
true or false:
 \((140 + 100) \div 20 = (140 \div 20) +
(100 \div 20)\)
 \( 240 \div (12 + 8) = (240 \div 12) + (240
\div 8)\)
 \( (300  60) \div 20 = (300 \div 20)  (60
\div 20)\)
 \((140 + 100) \div 20 = (140 \div 20) +
(100 \div 20)\)
Division is rightdistributive over addition and subtraction, for example, \((2 + 3) \div 5 = (2 \div 5) + (3 \div 5)\). The division symbol is to the right of the brackets. But it is not leftdistributive, for example, 10 Ã· (2 + 4) â‰ (10 Ã· 2) + (10 Ã· 4).
For example \((200 + 40) \div 20 = (200 \div 20) + (40 \div 20) = 10 + 2 = 12\), and
\((500 + 200  300) \div 50 = (500 \div 50) + (200 \div 50)  (300 \div 50)\)
 Evaluate each expression for
\(x = 2\) and \(x = 10\)
 \((10x^2 + 5x) \div 5 \)
 \( (10x^2 \div 5) + (5x \div 5) \)
 \( 2x^2 + x \)
 \((10x^2 + 5x) \div 5x \)
 \((10x^2 \div 5x) + (5x \div 5x) \)
 \(2x + 1 \)
 \((10x^2 + 5x) \div 5 \)
The distributive property of division can be expressed like this:
\((x + y) \div z = (x \div z) + (y \div z)\)
\((x  y) \div z = (x \div z)  (y \div z)\)

 Do not do any
calculations. Which of the following expressions do you
think will have the same value as
\((10x^2 +
20x  15) \div 5\), for \(x = 10\) as well as \(x
= 2\)?
\(2x^2 + 20x  15 \\ 10x^2 + 20x  3 \\ 2x^2 + 4x  3\)
 Do the necessary calculations to check
your answer.
 Do not do any
calculations. Which of the following expressions do you
think will have the same value as
\((10x^2 +
20x  15) \div 5\), for \(x = 10\) as well as \(x
= 2\)?
 Simplify:
 \((2x + 2y) \div 2 \)
 \((4x + 8y) \div 4 \)
 \( (20xy + 16x) \div 4x \)
 \( (42x  6) \div 6 \)
 \( (28x^4  7x^3 + x^2) \div x^2 \)
 \( (24x^2 + 16x) \div 8x\)
 \((30x^2  24x) \div 3x \)
 \((2x + 2y) \div 2 \)
 Simplify:
 \((9x^2 + xy) \div xy \)
 \((48a  30ab + 16ab^2) \div 2a \)
 \( (3a^3 +a^2) \div a^2 \)
 \((13a  17ab) \div a \)
 \((3a^2 + 5a^3) \div a \)
 \((39a^2b + 13ab + ab^2) \div ab \)
 \((9x^2 + xy) \div xy \)
The instruction \(72 \div 6\) may also be written as \(\frac{72}{6}\).
This notation, which looks just like the common fraction notation, is often used to indicate division.
Hence, instead of \( (10x^2 + 20x  15) \div 5 \) we may write \( \frac{10x^2 + 20x  15}{5}\).
Since \( (10x^2 + 20x  15) \div 5 \) is equivalent to \( (10x^2 \div 5) + (20x \div 5)  (15 \div 5) \),
\( \frac{10x^2 + 20x  15}{5}\) is equivalent to \( \frac{10x^2}{5} + \frac{20x}{5}  \frac{15}{5} \).
 Find a simpler
equivalent expression for each of the following expressions
(clearly, these expressions do not make sense if \(x =
0\)).
 \( \frac{16x^2  12x}{4x}\)
 \(\frac{16x^3  12x}{4x}\)
 \(\frac{16x^3  12x^2}{4x}\)
 \(\frac{16x^3  12x^2}{4x^2}\)
 \(\frac{16x^3  12x^2}{2x}\)
 \(\frac{16x^3  12x}{8x}\)
 \( \frac{16x^2  12x}{4x}\)
 In each case check whether the
statement is true for \(x = 10; ~x = 100;~ x = 5; ~x = 1\) and \(x = 2\).
 \(\frac{x^2}{x} = x\)
 \(\frac{x^3}{x} = x^2\)
 \(\frac{x^3}{x^2} = x\)
 \(\frac{5x^3}{5x} = 5x^2\)
 \(\frac{5x^3}{x} = 5^3\)
 \(\frac{5x}{x^2} = \frac{5}{x} \)
 \(\frac{x^2}{x} = x\)
 Explain why the equations below
are true:
 \(\frac{100x  5x^2}{5x} = 20x\) for all
values of \(x\) except \(x = 0\)
 \(\frac{15x^2  10x}{5x}\) is equivalent to \(3x 2\), excluding \(x = 0\).
 \(\frac{100x  5x^2}{5x} = 20x\) for all
values of \(x\) except \(x = 0\)
 Complete the table:
\(x\)
1,5
2,8
3,1
0,72
\(\frac{3x + 12}{3}\)
\(\frac{18x^2 + 6}{6}\)
\(\frac{5x^2 + 7x}{x}\)
(Hint: Simplify the expressions first to save yourself some work!)
 Simplify
each expression to the equivalent form requiring the fewest
operations.
 \( \frac{3a + a^2}{a}\)
 \( \frac{x^3 + 2x^2 x}{x}\)
 \( \frac{2a +12ab}{2a}\)
 \( \frac{12x^2 + 10x}{2x}\)
 \( \frac{21ab 14a^a}{7a}\)
 \( \frac{15a^2b +30ab^2}{5ab}\)
 \( \frac{7x^3 + 21x^2}{7x^2}\)
 \( \frac{3x^2+9x}{3x}\)
 \( \frac{3a + a^2}{a}\)
 Solve the equations.
 \( \frac{3x^2+15x}{3x} = 20\)
 \( \frac{30x  18x^2}{6x} = 2\)
 \( \frac{3x^2+15x}{3x} = 20\)
 Complete the table.
\(x\)
1,1
1,2
1,3
1,4
1,5
(a)
\( \frac{x^3 + 2x^2 x}{x}\)
(b)
\( \frac{7x^3 + 21x^2}{7x^2}\)
(c)
\(\frac{50x^2 + 5x}{5x}\)
 Simplify
the following expressions.
 \( \frac{3x(5x + 4) + 6 x(5x + 3)}{5x}\)
 \( \frac{14x^2 âˆ’ 28x}{7x} + \frac{24x  18x^2}{3x}\)
 \( \frac{3x(5x + 4) + 6 x(5x + 3)}{5x}\)
Products and squares of binomials
How can we obtain the expanded form of \( {\textbf(x + 2)(x + 3)}\)?
In order to expand \((x + 2)(x + 3)\), you can first keep \((x + 2)\) it is, and apply the distributive property:
\((x \textbf{+ 2})(x + 3) \\ = (x \textbf{+ 2})x + (x \textbf{+ 2})3 \\ = x^2 + 2x + 3x + 6 \\ = x^2 + 5x + 6\)
 Describe how can you check whether \((x + 2)(x + 3)\) is actually equivalent to \(x^2 + 5x + 6\).
To expand (x  y)(x + 3y) it can be written as (x  y)x + (x  y)3y and the two parts can then be expanded.
\((x  y)(x + 3y) \\ = (x  y)x + (x  y)3y \\ = x^2  xy + 3xy  3y^2 \\ = x^2 + 2xy  3y^2\)

Do some calculations to check whether \((x 
y)(x + 3y)\) and \(x^2 + 2xy  3y^2\) are
equivalent. Write the results of your calculations in the table
below.
\(x\)
\(y\)
 Expand each of these
expressions.
 \((x + 3)(x + 4)\)
 \((x + 3)(4  x)\)
 \( (x + 3)(x  5)\)
 \((2x^2 + 1)(3x  4)\)
 \((x + y)(x + 2y) \)
 \((a b)(2a + 3b)\)
 \( (k^2 + m)(k^2 + 2m)\)
 \((2x + 3)(2x  3)\)
 \((5x + 2)(5x  2) \)
 \((ax  by)(ax + by)\)
 \((x + 3)(x + 4)\)
 Expand
each of these expressions.
 \( (a+ b)(a + b)\)
 \( (a  b)(a  b)\)
 \((x + y)(x + y)\)
 \((x  y)(x  y)\)
 \((2a + 3b)(2a + 3b)\)
 \((2a  3b)(2a  3b)\)
 \( (5x +2y)(5x +2y)\)
 \( (5x  2y)(5x  2y)\)
 \((ax + b)(ax + b)\)
 \((ax  b)(ax  b)\)
 \( (a+ b)(a + b)\)

Can you guess the answer to each of the following questions
without working it out as you did in question 3? Try them out
and then check your answers.
Expand these expressions:
 \( (m + n)(m + n)\)
 \((m  n)(m  n)\)
 \((3x + 2y)( 3x + 2y)\)
 \( (3x  2y)( 3x  2y)\)
 \( (m + n)(m + n)\)
All the expressions in questions 4 and 5 are squares of binomials, for example \((ax + b)^2\) and \((ax  b)^2\)

Expand:
 \( (ax + b)^2\)
 \( (ax  b)^2\)
 \((2s + 5)^2\)
 \((2s  5)^2\)
 \((ax + by)^2\)
 \((ax  by)2\)
 \((2s + 5r)^2 \)
 \((2s  5r)^2 \)
 \( (ax + b)^2\)
 Expand and simplify.
 \( (4x + 3)(6x + 4) + (3x + 2)(8x + 5) \)
 \((4x + 3)(6x + 4)  (3x + 2)(8x + 5) \)
 \( (4x + 3)(6x + 4) + (3x + 2)(8x + 5) \)
Substitution into algebraic expressions

In question 2 you have to find the values of different expressions, for some given values of \(x\). Look carefully at the different expressions in the table. Do you think some of them may be equivalent?
Simplify the longer expression to check whether you end up with the shorter expression.
 Complete
the table.
\(x\)
13
13
2,5
10
(a)
\((2x + 3)(3x  5)\)
(b)
\(10x^2 + 5x  7 + 3x^2  4x  3\)
(c)
\(3(10x^2  5x + 2)  5x(6x  4)\)
(d)
\(13x^2 + x  10\)
(e)
\(6x^2  x  15\)
(f)
\(5x + 6\)
 Complete
this table.
\(x\)
1
2
3
4
(a)
\((2x + 3)(5x  3) + (10x + 9)(1  x)\)
(b)
\( \frac{9x^2 + 30x}{3x}\)
(c)
\(3x(10x  5)  5x(6x  4)\)
(d)
\(5x(4x + 3)  2x(7 + 13x) + 2x(3x + 2)\)
 Describe any patterns that you
observe in your answers for question 3.
 Complete
this table.
\(x\)
1,5
2,5
3,5
4,5
(a)
\((2x + 3)(5x  3) + (10x + 9)(1  x)\)
(b)
\( \frac{9x^2 + 30x}{3x}\)
(c)
\(3x(10x  5)  5x(6x  4)\)
(d)
\(5x(4x + 3)  2x(7 + 13x) + 2x(3x + 2)\)