Class 8 – Algebra

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Topic Sub Topic Online Practice Test
Algebra
  • Polynomials
  • Multiplication of algebraic expressions
  • Division of algebraic expressions
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Algebra
  • Special Products and identities
  • Factorization
  • Simplification of Algebraic Fractions
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Algebra
  • Simultaneous Linear equations
  • Linear Inequalities
  • Introduction to Quadratic equations
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Introduction: Algebra is a method of written calculations that help in reason about numbers. At the very outset, the student should realize that algebra is a skill. And like any skill — driving a car, baking cookies, and playing the guitar — it requires a lot of written practice.

The first thing to note is that in algebra use letters as well as numbers are used. But the letters represent numbers. We imitate the rules of arithmetic with letters, because we mean that the rule will be true for any numbers.

Here, for example is the rule for adding fractions:

The letters  mean: the numbers that are in the numerators.

The letter c means: the number in the denominator.

The rule means: “Whatever those numbers are, add the numerators and write their sum over the common denominator.”

Algebra tells us how to do any problem that is similar to the one given. That is one reason why we use letters.

The numbers are the numerical symbols or (constants) while the letters are called literal symbols (or) variables.

i)      The expression 3x – 9 is formed using the variable  and constants 3 and 9.

ii)     The expression 9xy – 5 is formed using the variables x, y  and constants 9 and 5.

A variable does not have a fixed value, it takes different values.

Algebraic Expression: A combination of terms separated by + (or) – signs are called as algebraic expressions.

Examples:

1.   4x – 9 is an algebraic expression which is made of two terms i.e 4x and 9.

2.   3x + 5y – 6 is an algebraic expression which is made up of three terms i.e 3x, 5y and 6.

Algebraic expressions are classified into various types based on number of terms in the expression.

Various types of algebraic expressions:

1.   Monomial: An algebraic expression containing only one term, is called a monomial.
Examples: 5x, 6y2, 3x2yz, -9 are all monomials.

2.   Binomial: An algebraic expression containing 2terms is called binomial.
Examples: (i) 5x + 9yis a binomial having 2 terms, namely 5x and 9y.
(ii) 2xy – 3 is a binomial having 2 terms, namely 2xyand 3.
(iii) a2b – 3b2c is a binomial having 2 terms, namely a2b and 3b2c.

3.   Trinomial: An algebraic expression containing 3terms is called a trinomial.
Example:  is a trinomial having3 terms, namely .

4.   Multinomial:  An algebraic expression containing more than 3 terms, is called a multinomial.A multinomial can have negative integers as exponents or fractional powers.

Example: i)  is a multinomial, having 4 terms namely , ,  and .

ii)  is a multinomial.              iii) is also a multinomial.

 

Factors of a Term: When numbers and literals are multiplied to form a product, then each quantity multiplied is called a factor of the product.

A constant factor is called a numerical factor while a variable factor is called a literal factor.

Thus,in , the numerical factor is -7 and the literal factors are

Constant Term: A term of the expression having no literal factors is called the constant term.
Thus, in the expression ,the constant term is -3

Clearly, the expression,   has no constant term.

Coefficient: Any factor of a term is called the coefficient of the product of other factors.
Examples:

(i)In 7xy, the numerical coefficient is 7 and literal coefficient is xy.Also, the coefficient of x is 7y and the coefficient of is 7x

(ii)In , the numerical coefficient is –9 and literal coefficient is
also, the coefficient of
the coefficient of
the coefficient of ,and so on

Like and unlike terms: An algebraic expression consists of two parts
i) Numerical factors                      ii) Literal or variable factors

Like terms: When the terms have same literal factors or variables, they are called like terms.

Example:

1) are like terms as they have common variable x.
2)  are like terms as they have common variable xy.

Unlike terms: When the terms have different literal factors or variables, they are called unlike terms

Example:  are unlike terms.

Polynomial: Polynomial is an algebraic expression that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.

  • Polynomials contains more than one term.
  • Polynomials are the sums of monomials.

a polynomial is a expression of the form
Where x is a variable, n is a positive integer, and  are constants.

Examples:

1.    is an expression but not a polynomial, since it contains a term namely  in which power of xis -1, which is not a non-negative integer.

2.    is an expression but not a polynomial, since it contains a term in which power of x is , which is not a non-negative integer

3.    is an expression but not a polynomial, as it contains a term in which power of z is , which is not a non-negative integer.

4.    is a polynomial.

 

Degree of a polynomial in one variable: The highest power of that occurs is called the degree of the polynomial.

Example:

1.    is a polynomial of degree 3.

2.    is a polynomial of degree 4.

Degree of polynomials in two or more variables: If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Example:

1.    is a polynomial in x and y, of degree 4.

2.    is a polynomial in and y of degree 6.

Classification of polynomials by degree:

Linear Polynomial: A polynomial of degree 1 is called a linear polynomial.

Example:  are all linear polynomials.

 

Quadratic Polynomial: A polynomial of degree 2 is called quadratic polynomial.

Example:  are both quadratic polynomials.

 

Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial

Example:  are both cubic polynomials.

 

Constant Polynomial: A polynomial having one term consisting of a constant only is a constant polynomial.

The degree of a constant polynomial is 0.

Thus, 6 is a constant polynomial.

 

Addition and subtraction of polynomials:
Addition of polynomials: To add two polynomials, combine their like terms and write the remining terms as they are.

Example:
Solution:

Example: Simplify
Solution:

Example: Given the length of a rectangle is  and breadth is  find algebraic expression for the perimeter of the rectangle.

Solution:

 

Subtraction of polynomials: To subtract one polynomial from another, change the subtraction sign to an addition sign and change the signs of all the terms in the polynomial being subtracted. Then add the resulting polynomial by combining like terms.

Example: Simplify
Solution:

 

Example: Simplify

Solution:

Example: The cost of a laptop is ` .Raghu paid `  to the store. Find an expression for the money he is yet to pay to the store.

Solution: Amount to be paid to the store = `

                                                                 = `

                     = `

http://images.clipartpanda.com/bloc-clipart-note-md.pngWhile adding algebraic expressions write each expression to be added in a separate row. While doing so we write like terms one below the other and add them. This method is called column method of addition.

 

Substitution: In a given expression, the process of replacing each variable by a given value is called substitution.

For finding the value of an expression for given values of the variables, we substitute the values of the variables as shown in the examples given below.

Example: If  find the value of .
Solution: Substituting  in the given expression, we get:

Example: Find the value of  when

Solution: Substituting  in the given expression, we get:

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To create a polynomial imagine carrying out the following steps:

1.   Start with a variable.

2.   Raise x to various integer powers, starting with the power 0 and ending with the power n (where n is a positive integer):

3.   Multiply each power of x by a coefficient. Let a3 denote the coefficient of x3, and so on.

4.   Add all the terms together.

The result is a polynomial.

Note that some of the coefficients could be zero so that some of the powers of x could be absent. Here is the formal definition of a polynomial

A polynomial is an expression of the form:

Where  is a variable,  is a positive integer and  are constants. The highest power of  that occurs is called the degree of the polynomial. The terms are usually written in order form highest power of x to lowest power.