ALGEBRA
Category : 6th Class
Learning Objective
The branch of mathematics which deals with numbers is called Arithmetic. Algebra can be considered as generalization of arithmetic, where we use letters in place of numbers, which allows to write rules and form in general way
VARIABLE
A symbol which takes various numerical values is called a variable.
CONSTANT
A symbol which takes a fixed numerical value is called a constant.
ALGEBRAIC EXPRESSIONS
When variables and constants are combined with the help of mathematical operations of addition, subtraction, multiplication and division, we get an algebraic expression. For example, 3x + 7, 15y - 23 are algebraic expression. 3x+7 is an algebraic expression in variable x, it is obtained by multiplying the variable x by constant 3 and then adding 7 to the product.
TERMS
Look at the expression (3x + 7). This is formed by first forming 3x as product of 3 and x and then adding 7 to the product. Similarly \[(10{{x}^{2}}+15)\] can be formed by first forming \[10{{x}^{2}}\] as product of 10, x and x and adding 15 to it. Such parts of an expression which are formed first and then added are called terms. Consider \[(9{{y}^{2}}-8x),\] here we can say that \[9{{y}^{2}}\] and - 8x are two terms of given expressions.
FACTORS OF A TERM
Now we know that an expression consist of terms. \[(9{{y}^{2}}-8x)\] has two terms \[9{{y}^{2}}\] and (-8x). The term \[9{{y}^{2}}\] is a product of 9, y and y. Here we say that 9, y and y are factors of term\[9{{y}^{2}}\]. A term is represented as product of its factors.
For term (-8.x), -8 and x are factors.
COEFFICIENTS OF A TERM
We know that any term of an expression can be expressed as product of its factors. These factors are numeric or variables. The numerical factor is called numerical coefficient or coefficient of the term.
In \[9{{y}^{2}}\], 9 is the coefficient of the term. It is also called coefficient ofy2. In - lO^z2, -10 is the coefficient of \[{{y}^{2}}{{z}^{2}}\].
If the coefficient of any term is + 1, we omit it.
\[1{{y}^{2}}\] can be written as y^{2}, 1xy is written as xy coefficient (-1) is indicated by minus (-) sign, \[(-1){{y}^{2}}\] is written as \[-{{y}^{2}},\,(-1)\,{{y}^{2}}{{z}^{2}}\] as \[-{{y}^{2}}-{{z}^{2}}\] etc.
as -y2, (-1) y^z2 as - y2 - z2 etc.
For example: In the following expressions identify the terms, factors and coefficients.
\[(4x+3y),\,3{{x}^{2}}-4x,\,3{{p}^{2}}q+7pq\,-8p{{q}^{2}}\]
Expression |
Terms |
Factors |
Coefficients |
\[4x+3y\] |
\[4x\] \[3y\] |
\[4,\,x\] \[3,y\] |
4 3 |
\[3x{{y}^{2}}-4x\] |
\[3x{{y}^{2}}\] \[-4x\] |
\[3,\,x,\,y,\,y\]\[-4,\,x\] |
3 -4 |
\[3{{p}^{2}}q+7pq\]\[-8p{{q}^{2}}\] |
\[3{{p}^{2}}q\] \[7pq\] \[-8p{{q}^{2}}\] |
\[3p,\,p,q\]\[7,\,p,\,q\] \[-8p,\,q,\,q\] |
3 7 -8 |
LIKE AND UNLIKE TERMS
In any algebraic expression, terms which have same variable(s) factor are called like terms. Terms which have different variable(s) factors are called unlike terms.
For example of expression \[3{{y}^{2}}+2x-2{{y}^{2}}+5,\] in this expression factors of \[3{{y}^{2}}\] are 3, y and y, factors of \[-2{{y}^{2}}\] are -2, y and y. Thus their variables factors are same so \[3{{y}^{2}}\] and \[-2{{y}^{2}}\] are like terms whereas \[3{{y}^{2}}\] and 2x are unlike terms because their variable factors are different, similarly 2x and 5 are also unlike terms.
TYPES OF ALGEBRAIC EXPRESSIONS
(i) An algebraic expression that has only one term is called a monomial.
For example:
\[x,\,4{{x}^{2}}y,\,-3{{p}^{2}}{{q}^{2}}\]etc.
(ii) An algebraic expression that contains two unlike terms is called a binomial.
For example:
\[x+y,\,3x+4y,\,x-10,\,-y-5\] etc.
(iii) An algebraic expression that contains three unlike terms is called a trinomial.
For example:
\[a+b+5,\,{{x}^{2}}-{{y}^{2}}+6,\,\,{{x}^{2}}y+x{{y}^{2}}+xy\] etc.
\[zy+10-y\] is not a trinomial as 2y and -y are like terms.
(iv) An algebraic expression that contains more than three unlike terms is called a polynomial.
For example:
\[{{x}^{3}}+4{{x}^{2}}+7x+3y+5,\,x{{y}^{2}}+7y+2x+3\] etc.
The degree of a Polynomial is the greatest of the exponents (indices) of its various terms.
For example:
(i) \[2+x+{{x}^{2}}+{{x}^{3}}\] is a polynomial of degree 3
(ii) \[3+2x+5{{x}^{2}}+{{x}^{4}}\] is a polynomial of degree 4
ALGEBRAIC EQUATIONS
An equation is a mathematical statement equating two quantities, e.g. 3x + 4 = 6x + 1, 7x + 2 = 11 + x, 12a + 9 = 8a + 4 etc.
SOLUTION OF EQUATION
It is the value of the unknown that balances an equation e.g.
(i) \[x+7=15\]
\[\therefore \,\,x=8\]
(ii) \[2x+3=7\]
\[2x=7-3\]
\[2x=4\Rightarrow \,x=\frac{4}{2}=2\]
ADDITION AND SUBTRACTION
Addition (or subtraction) is possible even if terms are like. Addition (or subtraction) of two unlike terms is not possible.
For example:
Add the following:
\[2x+3x=(2+3)\,x=5x;\]
\[5xy+7xy=(5+7)\,xy=12xy\]
\[8a-3a=(8-3)a=5a\] etc.
\[2{{x}^{2}}+4x+3-3{{x}^{2}}-5x+7\]
\[=(2-3){{x}^{2}}+(4-3)x+(3+7)\]
\[=-{{x}^{2}}+x+10\]
Hence, the sum (or difference) of several like terms is another like term whose coefficient is the sum (or difference) of the coefficients of several like terms.
MULTIPLICATION OF POLYNOMIALS
To multiply one polynomial with the other, (i) multiply each term of the polynomial by each term of the other, and (ii) then add the terms thus obtained.
For example:
(i)\[x(a+b)\,=xa+xb\]
[Distributive law of multiplication]
(ii) \[(x+a)\,(x+b)\,=x(x+b)\,+a(x+b)\]
\[={{x}^{2}}+xb+xa\,+ab\]
\[={{x}^{2}}+x(b+a)+ab\].
\[={{x}^{2}}+x(a+b)\,+ab\]
(iii) \[(3x+2)\,(x+4)\]
\[3x(x+4)\,+2(x+4)\]
\[=3{{x}^{2}}+12x+2x+8\]
\[=3{{x}^{2}}+14x+8\]
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