Algebra

ALGEBRAIC TERMS AND DEFINITIONS

Algebra simply entails the use of a combination of alphabets and numbers to formulate and solve mathematical problems. It could either be in form of an algebraic expression or an algebraic equation. The difference between the two is that while the former has no equal to (=) sign, the later has one.

Definition of terms

Product:

A product of two terms may be defined as the result obtained when they are multiplied together. The terms made reference to in this context could be real numbers, variables, unreal numbers, etc.

For instance, the product of a and y is given as: a*y=ay, in the same light, the product of 2*4=8

Factor:

A factor may be referred to as a term that can wholly be used to divide another term without any remainder. For instance, 2 is a factor of 22, because it can divide 22 without a remainder to yield 11. In the same light, 3x is a factor of 9xy+27xz, because 3x can wholly divide it without a remainder to yield 3y+9z.

Variable:

A variable can simply be described as an unknown in any expression or equation. They are usually represented with alphabets, for example; x, a, z, t, p, u, etc. These variables could stand alone or with each other as a single term.

Coefficient:

The coefficient of a variable is simply defined as the number that always comes before it. For instance, the coefficient of 7y is 7.

Constant:

In algebra, there exists a situation where numbers stand alone without alphabets after them and as such cannot be called coefficients. Such numbers are referred to as constants.

Power:

The power of a term can be defined as a number or variable to which the term is raised to. That is, if a term, say xy is raised to the power of 3, it means that xy must be multiplied by itself 3 times. This can be represented mathematically as:

   xy³=xy*xy*xy

As shown above, powers are usually represented in a superscript form; except when the power is 1, in which case, no number is written as its superscript. For instance, the power of a in the algebraic term 3ay²z³ is 1, that of y is 2 and that of z is 3.

Algebraic term:

This can be described as the combination of one or more variable(s) and its/their coefficient to yield a single term. Each variable in an algebraic term may or may not be raised to a power.Examples of algebraic terms include: 2x², 4xy, 9y, 8xt⁴p, etc.

NB: Generally, terms are simply units of any algebraic expression or equation. They could either be constants or algebraic terms.

Algebraic expression:

This may simply be defined as the combination of two or more algebraic terms with or without the inclusion of a constant.

Simple expression:

This is defined as an algebraic expression made up of algebraic terms which have only one variable. For example; 2x, 3b, 7x, 15c, etc.

Like terms:

This may be defined as algebraic terms that have variables that are exactly alike. That is, they have the exact same variable with the exact same powers respectively. For example, in the algebraic expression:

          12x²y – 7ay² + 3xyz -9x²y

12x²y and 9x²y are like terms.

Since the symbols a, b, c....x, y, z stand for numerical quantities we may apply the ordinary Arithmetic laws infusing them. In Arithmetic, 2*6+3*6 = 12+18 = 30. So in Algebra 2a+3a = 5a, 6a-2a = 4a. In Arithmetic 2*6 = 6*2, so in Algebra

a*b = b*a.

Also 2*4*6=4*6*2 = 6*2*4, so in Algebra

a*b*c = a*c*b = b*a*c = abc = acb = bca. If we then wish to add 3abc+2acb+7cab we should rearrange the terms thus,

3abc+2abc+7abc = 12abc. An important difference between the notation of Arithmetic and that of Algebra should be noted. In Arithmetic 34 means thirty-four or 3X10+4; in

Algebra ab means a*b.

Now, to see if you have being following so far, try the following simple exercises.

Exercises

i.      Find the values of the following:

2a—a

7x-3x

llx— 4x

x—x

3ab+5ab

ii.    What is the value of 8x when:

x = 2.

x = 4.

x= 1/2

x=-4.

iii.  What is the value of x/2 when

x = 4

x = 16

x = 5

x = 2/7

iv.    What is the number which is 2 greater than x?

v.      What is the number which is 3 less than x?

vi.    If an article costs x cents what is the cost of three articles?, of seven articles?, and of eleven articles?

vii.  Express x sq. ft. in sq. in.

viii.     Express x sq. in. in sq. ft.

ix.  Express x metres in (1) decimetres, (2) in centimetres,

(3) in millimetres, (4) in kilometres.

x.      4a+3a+6a-2a

xi.    7xy+6x²y² -3yx+4y²x²+3xy- 2x²y²

BODMAS

In algebra, BODMAS is of tremendous importance as it is used to determine the sequence of operations to be performed on an expression or equation to obtain its simplest possible form or even obtain a final answer.

BODMAS is an acronym which stands for:

B – Bracket

O – Of

D – Divide

M – Multiply

A – Addition

S – Subtraction

In any algebraic equation or expression where two or more of the operations defined above exist, the sequence “BODMAS” must be followed in executing said operations.

Brackets {()} are used to express multiplication. Anything (a number, sign or variable) that stands before a bracket multiplies everything within it. “OF” often means multiplication and is usually executed before the traditional multiplication sign (*).

The other operations are very conventional and will be explained with example given below.

Examples:

i.      3a+(4a-2a) = 3a+6a = 9a

ii.    15x-(6x+3x) = 15x-9x = 6x

iii.  3b-(2a+4a) = 3b-6a

iv.    6a-(4a+2a) = 6a-6a = 0

v.      6+(x-2)-(3+4x)+(6x+1) = 6a+2b

Now, try these?

Exercises

i.      (3x-2)-(4x+5)+2(x+7)=

ii.    3(a+b+c)-(b+a-c)-(2c-2a-b)=

iii.  2(3x+12)+3(x+4)-(8x-12) =

iv.    3{x-(2x-6x)} =

v.      X+{2x+3(x+2x)} =

vi.  3x²+x(x+3)+x² =

  

FACTORS

A factor may be defined as a term (number, variable, a combination of both or an expression) that can wholly be obtained from a bigger/more complex expression. A factor is usually able to divide the parent expression without any remainder. For instance, 2x is a factor of 12x²y+8xy², this is because when it completely divides 12x²y+8xy², it yields 6xy+4y² without any remainder. The expression 12x²y+8xy²can thus be written as:

          12x²y+8xy² = 2x(6xy+4y²)

The simply act of rewriting 12x²y+8xy² as 2x(6xy+4y²) is referred to as factorization.

FACTORIZATION OF SIMPLE ALGEBRAIC EXPRESSIONS

Factorization of algebraic expressions entails the breaking down of a complex algebraic expression to a simpler form by bringing out the common factors of all the terms in the expression and enclosing what is left in a bracket. Practical examples of this phenomenon are shown below:

Examples:

i.      3xy-2y

= y(3x-2)

ii.    14x²za+7x³ya

= 7x²a(2z+xy)

iii.  32ab²c+16a³b²c²-64a²b³c

= 16ab²c(2c+a²c-4ab)

It is also noteworthy that in some cases, there is need to remove the bracket before simplification via factorization.

 Examples:

i.      X(a+3b) + b(2x-3a)

= ax+3bx+2bx-3ab

= ax+5bx-3ab

= x(a+5b) – 3ab

ii.    2x + 3(x+2y-4z)

= 2x+3x+6y-12z

= 5x+6y(y-2z)

Generally, in factorization, the highest common factor(HCF) of the terms is taken out and is used to divide them(the terms) with the remainders placed inside the brackets. Example:

i.      2ab+4a²b-6ab²

= 2ab(1-2a-3b)

ii.    (a-x)(3a+2x) – (a-x)²

= (a-x)[(3a+2x) – (a-x)]

= (a-x)[3a+2x-a+x]

= (a-x)(2a+3x)

Factorization by rearrangement of terms:

In most factorization cases, the terms that are alike(that is, have common factors) may not be together. In such a case, it is ideal to rearrange such as expression before further factorization is carried out. In this case, after rearranging the terms(if need be), the like terms are taken out and the remainders are put in brackets as usual. In cases where the remainders are alike, they can be regrouped, thus factorizing further. The examples below will help for better understanding of this phenomenon.

Examples:

i.      2dx+2dy+cx+cy

ii.    ab+b²-ay-yb

iii.  4mb+4bx+mx+mn

iv.    5ax-5bx-a+b

v.      m²-mq-n²-nq

Solutions:

i.      2dx+2dy+cx+cy

= 2d(x+y) + c(x+y)

= (x+y)(2d+c)

ii.    ab+b²-ay-yb

= b(a+b) – y(a+b)

= (a+b)(b-y)

iii.  4mb+4bx+mx+mn

Rearranging; 4mb+4bx+mn+nx

= 4b(m+x) + n(m+x)

= (m+x)(4b+n)

iv.    5ax-5bx-a+b

= 5x(a-b) – (a-b)

= (a-b)(5x-1)

The fifth example brings us to another method of resolving factorizable expressions, which is the factorization of expressions which are difference of two squares.

As the name implies, the difference of two squares is simply an expression that contains two terms both completely squared and a subtraction sign between them. An example of this is   “x²-y²”. Normally, when we are given an expression of this nature, we simply solve it thus:

              x²-y² = (x+y)(x-y)

The result above can be verified by expanding the result. Thus, by this definition, example (v) can be solved thus:

v.      m²-mq-n²-nq

= m²-n²-mq-nq

= (m²-n²)-q(m+n)

= [(m+n)(m-n)]-q(m+n)

= (m+n)[(m-n)-q]

= (m+n)[m-n-q]

     SIMPLE EQUATIONS

Factors:

As previously defined in the preceding topic, factors are simply terms (numbers, variables, a mixture of both or a simpler expression) that can wholly divide a more complex expression without a remainder. It is noteworthy that in picking factors, one must seek out the HCF (highest common factor). This is so as to avoid factorizing even further, thus eliminating unnecessary complications from our work.

Equation:

An equation is a group of algebraic terms with an equal to (=) sign. It differs greatly from expressions. Examples of equations include:   x+y=2, y=3, 2x²+5b-3abc=2x

Root of an equation:

The root(s) of an equation maybe defined as the solution(s) obtained by completely solving an algebraic equation. For our study, we shall make reference to linear equations and equations of higher degrees that will yield only one root. Quadratic equations usually result in 2 roots, etc.

Solutions to simple equations

Examples:

1.      Find the roots of the equations below:

i.      12x²-3x = x

ii.    7a²b+14ab = 21ab

iii.  X³y²-3xy²+x³y² = 2(xy+x³y²)

Solutions:

i.      12x²-3x = x

12x² = 3x+x (collecting like terms)

12x² = 4x (Divide through by 12x)

12x(x) = 4x

X = 4x/12x = 1/3

Thus, x = 1/3

ii.    7a²b+14ab = 21ab

7a²b= 21ab-14ab (collecting like terms)

7a²b = 7ab (dividing through by 7ab)

(7ab)a = 7ab

a = 7ab/7ab = 1

Thus, a = 1

iii.  X³y²-3xy²+x³y² = 2(xy+x³y²)

X³y²-3xy²+x³y² = 2xy+2x³y²

Collecting like terms,

X³y²+x³y²-2x³y²-3xy²= 2xy

-3xy² = 2xy

Dividing through by -3xy

(-3xy)y = 2xy

Y = 2xy/(-3xy) = -2/3

Thus, y = -2/3

2.      3x+2 = 8

Collecting like terms;

3x = 8-2

3x = 6

Dividing through by 3;

X = 6/3

Thus, x = 2

3.      5y = 8+3y

Collecting like terms;

5y-3y = 8

2y = 8

Y = 8/2 (Dividing through by 2)

Thus, y = 4

3(4y-7) – 4(4y-1) = 6y-37 (Expanding;)

12y-21-16y+4 = 6y-37

Collecting like terms;

12y-16y-6y = -37+21-4

-10y = -20

Dividing through by -10

Y = -20/-10

Thus, y = 2