Beyond on Symmetrical Algebraic Expressions

of 9 O level Math Note KLAng June 2012

A first introduction to symmetrical algebraic expressions

Background information:

In mathematics competitions at all level, from secondary school to International

Mathematical Olympiad(IMO), problems related to symmetrical properties are a common

place. This is a first attempt to introduce this topic to lower secondary students who are keen

to take part in any of these competitions. So, lets enjoy.

What is a symmetrical expression?

Give that a, b, c are constants; x, y, z are variables,

ax, by, cz are similar terms; axy, byz, czx are similar terms; ax2y, by

2z, cz

2x are similar terms.

ax + ay, degree 1 symmetrical expression of x, y variables.

ax2 + bxy + ay

2, degree 2 symmetrical expression of x, y variables.

ax3 + bx

2y + bxy

2 + ay

3, degree 3 symmetrical expression of x, y variables.

Golden property of symmetrical expressions:

For two symmetrical expressions, the operations of sum, difference, product, and division, yield

a symmetrical expression.

The degree of an expression is a concept similar to that of a polynomial.

For these variables, x, y, z , these are degree 1 expressions: 3x, 2y, x + y + z.

These are degree 2 expressions: 4x2, x

2 + y

2, ( x + y )

2 , xy, xz, yz.

These are degree 3 expressions: x3, x

3 + z

3, ( x + y )

3 , xyz, x

2z, yx

2+yz

2+zx

2.

This is a degee 1 symmetrical expression: x + y + z. What it means is that if x is exchanged with

either y or z, the expression remains the same. xy + yz + zx is a degree 2 symmetrical

expression. xyz is a degree 3 symmetrical expression.

So, x2 + 2xy + y

2 is a degree 2 symmetrical expression of variable x, y.


Lets begins with a few worked examples:

Example 1:

Expand 5yx .

yx is a degree 1 symmetrical expression of variable x , y .

yxyxyxyxyxyx 5 the product is still a symmetrical expression.

5yx is a degree 5 symmetrical expression of variable x , y .

Degree 5 variables, 55 yx , xyyx 44 , 2323 xyyx

Let 232344555 xyyxcxyyxbyxayx , where a, b, c are constants.

Equating 5x term, 1a

When 1x , 1y , cb 2222 5 cb15

When 2x , 1y , cb 414311 5 cb 2715

Solve the two equations, 5b , 10c

232344555 101055 xyyxxyyxyxyx

We can use Binomial Theorem to show that,

543223455 510105 yxyyxyxyxxyx

Example 2:

Expand 3zyx .

3zyx is a degree 3 symmetrical expression of variable x , y , z .

Degree 3 variables, 333 zyx , yzxyzxxzzyyx 222222 , xyz . It may take awhile for a learner to list these variables correctly.

Let xyzcyzxyzxxzzyyxbzyxazyx 2222223333 , where a, b, c are constants.

Equating 3x term, 1a

When 1x , 1y , 0z , b222 3 3b


When 1x , 1y , 1z , cb 211 3 6c

xyzyzxyzxxzzyyxzyxzyx 63 2222223333

Example 3:

Simplify

zyx

xyzyxxzzy

.

yxxzzy is a degree 3 symmetrical expression of variable x , y , z .

xyz is a degree 3 symmetrical expression of variable x , y , z .

xyzyxxzzy is also a degree 3 symmetrical expression of variable x , y , z .

zyx is a degree 1 symmetrical expression of variable x , y , z .

When zyx , 0 xyzyxxzzy . zyx is a factor of the expression.

zyx

xyzyxxzzy

is a degree 2 symmetrical expression of variable x , y , z .

Degree 2 variables, 222 zyx , zxyzxy .

Let zxyzxybzyxa

zyx

xyzyxxzzy

222 , where a, b are constants.

When 1x , 1y , 0z , ba 22

2 ba 21 ---------- (1)

When 1x , 1y , 1z , ba 333

9 ba 1 ---------- (2)

Solving (1) and (2) simultaneously, we get 0a , 1b

zxyzxyzyx

xyzyxxzzy


Example 4:

Simplify

xyz

xzzyyxzxyzxy 2222222

.

2zxyzxy is a degree 4 symmetrical expression of variable x , y , z .

222222 xzzyyx is also a degree 4 symmetrical expression of variable x , y , z .

xyz is a degree 3 symmetrical expression of variable x , y , z .

zyx is a degree 1 symmetrical expression of variable x , y , z .

When 0x , 02222222 xzzyyxzxyzxy . So, by symmetry, xyz is a factor of the expression.

zyx

xyzyxxzzy

is a degree 1 symmetrical expression of variable x , y , z .

Let zyxa

xyz

xzzyyxzxyzxy

2222222

where a is a constant.

When 1x , 1y , 1z , a31

6 2a

zyx

xyz

xzzyyxzxyzxy

2

2222222

Example 5:

Factorise 555 yxyx .

When yx , 0555 yyyy . So, yx is a factor of this expression.

When 0x , 055 yy . So, x is a factor of this expression. Since this is a symmetrical expression, y is also factor of the expression.

Let xyqyxpyxxyyxyx 22555 where p, q are constants.


When 1x , 1y , qp 2211125 152 qp ---------- (3)

When 2x , 1y , qp 2512121 5 1525 qp ---------- (4)

Solving (3) and (4), we get 5p , 5q .

xyyxyxxyyxyx 22555 5

Example 6:

Factorise xyzzyx 3333 .

After some trials and errors, we can find that zyx is a factor of this expression.

Let zxyzxyqzyxpzyxxyzzyx 222333 3 where p, q are constants.

When 1x , 0y , 0z , p11 1p

When 1x , 1y , 0z , q 222 1q

zxyzxyzyxzyxxyzzyx 222333 3

It remains a challenge in factorising an algebraic expression. Guessing a correct factor of an

expression is still very much dependent on the past experience.

What is a rotational symmetrical expression?

The expression xzzyyx 222 is not symmetrical. If x is exchanged with y, the expression becomes yzzxxy 222 that is not the same as the original expression. However, if x is changed with y, y is exchanged with z and z is exchanged with x, then, the expression remains

the same. xzzyyx 222 is called a rotational symmetrical expression.

Some of these rotational symmetrical expressions:

333 xzzyyx , yxzxzyzyx , uwwzzyyx 2222 .

Golden property of rotational symmetrical expressions:

For two rotational symmetrical expressions, the operations of sum, difference, product, and

division, yield another rotational symmetrical expression.


For a degree 3 and above variable of x , y , z , there is a different treatment between a non-

rotational and rotational symmetry.

For a non-rotational symmetry, we write,

cxyzyzxyzxxzzyyxbzyxa 222222333

As for a rotational symmetry, it is modified,

dxyzyzxyzxcxzzyyxbzyxa 222222333 .

Each group is rotationally the same. Of course, a rotational symmetry group is also a sub-

group in the symmetry group.

Lets study a few examples:

Example 7:

Expand baaccb .

baaccb is a degree 3 rotational symmetrical expression of variable a , b , c .

Let abcsbcabcaraccbbaqcbapbaaccb 222222333 , where p, q, r, s are constants.

Since baaccb does not have 3a term, 0p .

When 1a , 1b , 0c , rq 0 rq ---------- (5)

When 1a , 1b , 0c , rq 2 ---------- (6)

Solving (5) and (6), we get 1r , 1q .

When 1a , 1b , 1c , srq 330 0s

accbbabcabcabaaccb 222222

Example 8:

Simplify

accbbaaccbba

333

.

333 accbba is a degree 3 rotational symmetrical expression of variable a , b , c .


accbba is a degree 3 rotational symmetrical expression of variable a , b , c .

When ba , 0333 accbba . ba is a factor of the expression. By

rotational symmetry, accb are also factors of the expression.

When

accbbaaccbba

333

, we get a degree 0 expression, which is a constant.

Let

k

accbba

accbba

333

, where k a constant.

When 2a , 1b , 0c , 2

6

k 3k

3

333

accbba

accbba

Example 9:

Factorise 555 accbba .

When ba , 0555 accbba . So, ba is a factor of this expression. Since this is a rotational symmetrical expression, cb and ac are also factor of the expression.

Let cbafaccbbaaccbba ,,555 where cbaf ,, is a degree 2 rotational symmetrical expression.

cabcabqcbapcbaf 222,, , the degree 2 rotational symmetry is the same as the symmetry ones.

When 1a , 1b , 0c , qp 21121125 152 qp ---------- (7)

When 2a , 1b , 1c , qp 6123210 356 qp --------- (8)

Solving (7) and (8), we get 5p , 5q .

cabcabcbaaccbbaaccbba 222555 5


The next few examples will mainly deal with fractional expressions.

Example 10:

Simplify yzxz

z

xyzy

y

zxyx

x

333

.

The given fraction is a rotational symmetry. After some rearrangement, we get

yzxzz

xyzy

y

zxyx

x

333

xzzyz

zyyx

y

xzyx

x

333

xzzyyx

yxzxzyzyx

333

When yx , 033 yzyzyy , therefore yx , zy , xz are factors of the numerator.

Let zyxxzzyyxpyxzxzyzyx 333

When 2x , 1y , 0z , 321122 3 p 1p

zyxxzzyyxyxzxzyzyx 333

zyx

yzxz

z

xyzy

y

zxyx

x

333

Example 11:

Given that cb

ax

,

ac

by

,

ba

cz

, prove that 1

111

z

z

y

y

x

x.

When each given fraction is changed from x to y to z, the others are changed from a to b to c.

The given fractions are rotational symmetries.


cba

a

cb

acb

a

x

x

11

By rotational symmetry, we get

1

cba

cba

bac

c

acb

b

cba

a

In conclusion:

In this article, I have introduced the basic properties of symmetrical expressions in relation to

expansion, simplification, and factorization, through the examples. Needless to say, one of

the most importance as well as challenge is in factorization. Guess and check, with the

demonstrated approaches, are served as leads in each factorisation. As symmetry has a

particularly important place in competition mathematics, I will most likely be revisiting this

topic again soon.

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Beyond on Symmetrical Algebraic Expressions