## Algebra

Category : 6th Class

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

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$-8$.

•                  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.

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