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ALGEBRA

 

ALGEBRA

The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.

 

Introduction to Algebra

•                   Variable: A letter symbol which can take various numerical values is called variable or literal.

Example: \[x,\,\,y,\,\,z\]etc.

 

•                   Constant: A symbol which can take a fixed numerical value is called a constant.

Example: \[-1,\,\,\frac{1}{2},\,\,2,\,\,4,\,\,3,\,\,5\]etc.

 

•                   Term: Numericals or literals or their combinations by operation of multiplication are called terms.

Example: \[5{{x}^{2}},\,\,7x,\,\,\frac{x}{7},\,\,\frac{{{y}^{2}}}{9},\,\,\frac{5}{2}x\]etc.

 

•                   Constant Term:  A term of an expression having no literal is called a constant term.

Example: \[4,\,\,\frac{-1}{2},\,\,\frac{7}{4}\]etc.

 

•                Algebraic expression: A combination of constant and variable connected by mathematical operations \[(+,\,\,-,\,\,\times ,\,\,\div )\] is called an algebraic expression.

Example: \[2a+3,\text{ }2a+3b,\text{ }7n+4,-p+-r+2\]etc.

 

Types of Algebraic Expressions;-

•                   Monomial: An expression containing only one term is called a monomial.

Example: \[7x,\,\,-11{{a}^{2}}{{b}^{2}},\,\,\frac{-7}{5}\]etc.

 

•                   Binomial: An expression containing two terms is called a binomial.

Example:\[2x+3,\text{ }6x-5y\] etc.

 

•                  Trinomial: An expression containing only three terms is called trinomial.

Example: \[2x+3y-\frac{5}{2},\frac{a}{2}-\frac{b}{3}+4\]etc.

•                   Multinomial: An expression containing only two or more terms is called a multinomial.

Example:\[2x+3y-2,\text{ }7+6x+3y,\text{ }3\sqrt{x}+4\]etc.

 

•                   Degree of a monomial: The degree of a monomial is the sum of the indices (power) in each of its variables.

Example: The degree of \[6{{x}^{2}}y\,\,{{z}^{3}}\]is\[2+1+3=6\]

 

•                   Degree of polynomial: The highest power of terms in a polynomial is called the degree of a polynomial.

Example: Degree of\[6{{x}^{2}}-5{{x}^{2}}2x-3\] is ' 3'. The degree of\[6{{x}^{6}}+5{{x}^{5}}\text{ }{{y}^{7}}+9\] is 12.

 

•                   Zero polynomial: If all the coefficients in a polynomial are zeros, then it is called a zero polynomial.

 

•                   Zero of the polynomial: The number for which the value of a polynomial is zero, is called zero of the polynomial.

Example: \[x=1,\]is called a zero of\[x-1\].

 

•                   Polynomial: An expression containing one or more terms with positive integral powers is called a polynomial.

Example:\[8{{x}^{2}}+2x+3,\]         \[2p-3q+\frac{5}{2}r\]etc.

•                    Factors: In a product, each of the literals or numerical values is called a factor of the product.

Example: \[15=3\times 5,\]where 3, 5 are called the factors of 15.

\[10xy=2\times 5\times x\times y,\]where\[2,\,\,5,\,\,x,\,\,y\]are called the factors of\[10\,\,xy\].

 

•                  Coefficient: In a product containing two or more than two factors, each factor is called coefficient of the product of other factors.

Example: In 6xy, 6 is called numerical coefficient of \['xy'\] and \['x'\] is the literal coefficient of \['7y'\] and \['y'\] is the literal coefficient of\[7x\].

 

Note: Degree of zero polynomial is not defined but some of the famous mathematicians claim the degree of zero polynomial is defined as \[-1\] or\[-\infty \].

 

•                   Substitutions: The method of replacing numerical values in the place of literal numbers is called substitution.

Example: The value of \[9x\] at \[x=4\]is\[9\times 4=36\]

 

Operation of Algebraic Expression

•                  Addition: Addition of algebraic expressions means adding the like terms of the expressions.

Note: Unlike terms cannot be combined or added.

Example: \[2x+5x=7x,\,\,8xy+9xy=17xy\]etc.

•                        Combining the coefficients of like terms of an expression through addition or subtraction is called simplication of an algebraic expression.

 

•                        There are two methods of adding algebraic expression. They are
•            Horizontal method
•            Vertical method

 

Horizontal method  

In this method, like terms should be added and unlike terms should be written separately by using associative law of addition.

Example:        \[7x+4y\]and \[3x-5y\]

Solution:        \[7x+4y+3x-5y\]

                                    \[=7x+3x+4y-5y\]

                                    \[=10x-y\]

 

Vertical method: following the steps.

•           In this method the expressions to be added, are written one below the other.
•           The like terms of each type are placed in separate columns.
•           The sum will be written below that column.

 

Example:       

                       

 

Subtraction of Algebraic Expressions:

The additive inverse of a number

     The additive inverse of any number is obtained by a simple change of its sign, so additive inverse of a number is also called the negative of that number.

Example: Additive inverse of \[8\] is\[-8\].

 

Additive Inverse of Expression

•                  The additive inverse or the negative of an expression is obtained by replacing each term of the expression by its additive inverse.

 

Example: 1 Additive inverse of\[-~9x\]is\[9x\].

 

Example: 2   Subtract \[16x-5y\]from \[7x+4y\]

Solution:          \[(7x+4y)-(16x-5y)\]

                        \[=7x+4y-16x+5y\]

                        \[=-9x+9y\]

 

Example: 3   Subtract \[3\times 2-5x-4\]from \[5{{x}^{2}}+6x+8\]

Solution: \[(5{{x}^{2}}+6x+8)-(3{{x}^{2}}-5x-4)\]additive inverse of \[\left( 3{{x}^{2}}-5x-4 \right)\]is \[-3{{x}^{2}}+5x-4\]

then, \[5{{x}^{2}}+6x+8-3{{x}^{2}}+5x-4\]

\[=\text{ }5{{x}^{2}}-3{{x}^{2}}+6x+5x+8-4\]

\[=2{{x}^{2}}+11x+4\]

 

Subtraction can also be done in two ways;

•                   Horizontal method:

Example: \[\left( x+y \right)-\left( 2x+3y \right)\]

\[=x+y-2x-3y\]

\[=-x-2y\]

 

•          Vertical method:

Example:

 

 

Multiplication of Algebraic Expression

•                   First remember the Rule of Exponent.

\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\] (a is any variable and m, n are positive integers.)

e.g. \[{{x}^{6}}\times {{x}^{7}}={{x}^{6+7}}={{x}^{13}}\]

 

Multiplication of monomials

•                   Product of two monomials = (Product of their numerical coefficients) \[\times \] (product of their variables).

Example:          \[6xy\]and \[-3{{x}^{2}}y\]

Solution:          \[(6xy)\times (-3{{x}^{2}}y)\]

                        \[=\{6\times -3\}\times ({{x}^{2}}y\times xy)\]

                        \[=-18{{x}^{3}}{{y}^{2}}\]

Multiplication of a Binomial and a monomial;

•                   Use distributive properly \[x\,\,(y+z)=xy+xz.\]

Example: \[2x\left( 3y+z \right)=2x\times 3y+2xz\]

\[=6xy+2xz\]

 

Multiplication can also done in two ways

•                   Horizontal method:

Example: \[2x\left( 3x+5z \right)\]

                  \[=2x\times 3x+2x\times 5z\]

                  \[=6{{x}^{2}}+10xz\]

•                   Column method:

 

Multiplication of two binomials

•                    Suppose two binomials \[(x+y)\] and \[(a+b)\] and using the distributive law of multiplication over addition twice.

\[\left( x+y \right)\times \left( a+b \right)=x\left( a+b \right)+y\left( a+b \right)\]

\[=\left( x\times a+x\times b \right)+\left( y\times a+y\times b \right)\]

\[=ax+bx+ay+by\]

Example: \[(3x-5y)\times (7x+4y)\]

\[=3x(7x+4y)-5y(7x+4y)\]

\[=(2x\times 7x+3x\times 4y)+(-5y\times 7x+4y\times -5y)\]

\[=21{{x}^{2}}+12xy-35xy-20{{y}^{2}}\]

\[=21{{x}^{2}}-23xy-20{{y}^{2}}\]

 

Division of Algebraic Expression

•                 First remember the rule of exponent \[{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}\](Where \[m>n\] and m and n are positive integers and a is variable)

Example: Divide \[{{x}^{10}}\] by \[{{x}^{5}}\]

Solution: \[{{x}^{10}}\div {{x}^{5}}={{x}^{10-5}}={{x}^{5}}\]

 

Division of monomials

•                   Rule: Quotient of two monomials

= (quotient of their numerical coefficients) \[\times \](quotient of their variables)

Example: \[~38{{x}^{2}}{{y}^{2}}\]by \[19xy\]

Solution: \[38{{x}^{2}}{{y}^{2}}\div 19xy\]

                        \[=\frac{38}{19}\times \frac{{{x}^{2}}{{y}^{2}}}{xy}\]

                        \[=2\times {{x}^{2-1}}.{{y}^{2-1}}\]

                        \[=2xy\]

 

•                    Division of a polynomial by a monomial

Example: \[12{{x}^{4}}-6{{x}^{2}}+3x\] by \[3x\]

Solution: \[\left( 12{{x}^{4}}-6{{x}^{2}}+3x \right)\div 3x\]

\[=\frac{12{{x}^{4}}}{3x}-\frac{6{{x}^{2}}}{3x}+\frac{3x}{3x}\]

\[4{{x}^{3}}-2x+1\]

 

Division of a polynomial by a polynomial

•                   Arrange the terms of the dividend and divisor in decreasing order of powers keeping zero for missing term.
•                   Divide the first term of the dividend by the first term of the divisor and write the result as the first term of the quotient.
•                  Multiply the entire divisor by this first term of the quotient and put the product under the dividend, keeping like terms under each other.
•                   Subtract the product from the dividend and bring down the rest of the dividend.
•                   Step 4 gives use the new dividend, repeat steps 1 to 4.
•                   Continue the process till the degree of the remainder becomes not defined or less than that of the divisor.

Example: 1 \[\left( {{x}^{2}}+2x+1 \right)\] by \[\left( x+1 \right)\]

Example: 2 Divide \[{{x}^{3}}+8\]by \[x+2\]

Special Products

•                   Special products and its proof:

(i) \[{{\left( a+b \right)}^{2}}+{{a}^{2}}+2ab+{{b}^{2}}\]

            LHS      \[={{\left( a+b \right)}^{2}}\]

                        \[=\left( a+b \right)\left( a+b \right)\]

                        \[=a\left( a+b \right)+b\left( a+b \right)\]

                        \[={{a}^{2}}+ab+ab+{{b}^{2}}\]

                        \[={{a}^{2}}+2ab+{{b}^{2}}\]

LHS = RHS

(ii) \[{{\left( a+b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]

LHS \[=\left( a-b \right)\left( a-b \right)\]

            \[=a\left( a-b \right)-b\left( a-b \right)\]

            \[={{a}^{2}}-ab-ab+{{b}^{2}}\]

            \[={{a}^{2}}-2ab+{{b}^{2}}\]

LHS = RHS

(iii) \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]

LHS \[=\left( a+b \right)\left( a-b \right)\]

            \[=a\left( a-b \right)+b\left( a-b \right)\]

            \[={{a}^{2}}-ab+ab-{{b}^{2}}\]

            \[={{a}^{2}}-{{b}^{2}}\]

LHS = RHS

 

Example: 1   If \[x+\frac{1}{x}=5,\]find \[{{x}^{2}}+\frac{1}{{{x}^{2}}}\]

Solution:       \[x+\frac{1}{x}=5\]

Squaring both sides

\[{{\left( x+\frac{1}{x} \right)}^{2}}={{\left( 5 \right)}^{2}}\]

\[={{x}^{2}}+\frac{1}{{{x}^{2}}}+2\times x\times \frac{1}{x}=25\]

\[={{x}^{2}}+\frac{1}{{{x}^{2}}}+2=25\]

\[={{x}^{2}}+\frac{1}{x}=25-2\]

\[\therefore \]\[{{x}^{2}}+\frac{1}{{{x}^{2}}}=23\]

 

Example: 2   Expand \[{{(x-3y)}^{2}}\]

Solution:      \[\left( x-3y \right)\,\,\left( x-3y \right)\]

                        \[={{x}^{2}}-2\times 3y\times x+{{\left( 3y \right)}^{2}}\]

                        \[={{x}^{2}}-6xy+9{{y}^{2}}\]

 

Linear equation in one variable;

Equation:

•                    If two numerical expressions are joined or connected by the symbol "is equal to (=)", then the combination is called as an equation.

 

LHS and RHS Notations

•               The sign of equality '=' in an equation divides it into two sides namely, the left hand side and the right hand side, written as LHS and RHS respectively.

 

Example:     \[7x+2=5x+3\]

Here, \[7x+2=\]LHS and \[5x+3=\]RHS

 

Linear Equation in one variable

•                  Linear equation which involves one variable is called linear equation in one variable or simple equation.

Example: 1   \[4x-3=x+5\]

Example: 2   \[y+8=11\]

 

•                   General form of linear equation is\[ax+b=0\], where \[(a\ne 0)\] and \[a,\,\,b\]are real numbers.

 

Solution of an Equation

•                  The values of variables which satisfies (LHS = RHS) the equation is called the solution or root of an equation.

Example:     \[2x+10=4\]

Here, LHS \[=2x+10,\]RHS\[=14\]

Now, above equation is true only when \[x=2\]i.e. \[x=2\]

\[=LHS=2\times 2+10=14\]

\[RHS=14\]

\[\therefore \] LHS = RHS

\[\therefore \] Root of \[2x+10=14\] is 2.

Type of finding the solutions of linear equations

•                    Transposition: Any term of an equation may be taken to the other side with a change in its sign. This process is called transposition.

Example: (i)    \[x+5=7\]

Solution:         \[x=7-5\]

\[\therefore \]\[x=2\]

 

Example: (ii)                \[\frac{x}{8}=5\]

Solution:          \[x=5\times 8\]

\[x=40\]

 

Example: (iii)   \[2x=40\]

Solution:           \[2x40\]

                                    \[x=\cancel{\frac{40}{2}}\]

                                    \[\therefore \]\[x=20\]

 

Example: (iv)    \[3x+2=x-2\]

Solution:        \[2x=-4\]

                                    \[x=\cancel{\frac{4}{2}}=-2\]

                                    \[\therefore \]\[x=-2\]

 

Example: The sum of two numbers is 100 and their difference is 10. Find the numbers.

Solution: Let the numbers be\[x\], \[100-x\]

                        \[x-(100-x)=10\]

                        \[x-100+x=10\]

                        \[2x=110\]        

                        \[x=55\]

Then the numbers are 55 and 45.