E-learning Class 8 Maths Free Tutorial – Algebraic Expressions and Identities



introduction algebra is a branch of mathematics that involves letters for numbers the numbers are the constants the letters used are called of variables an expression contains the combination of constants variables together with the arithmetic operators when two expressions are equated together we get an equation some equations are true for all the values and such equations are called identities objectives at the end of this lesson you'll be able to identify the algebraic expressions solve degree of polynomial solve monomials binomials and polynomials identify like and unlike terms solve addition subtraction and multiplication of the algebraic expressions describe identity and standard identities what are expressions the price of a chocolate is rupees five and the price of a pen is rupees ten what is the total price of three chocolates and ten pens to solve their problem we form an expression as cost of one pen is equal to rupees x cost of one chocolate is equal to rupees y cost of three chocolates is equal to rupees three y cost of ten pen is equal to rupees ten x total cost is equal to three x + 10 y which is equal to 3 into 5 plus 10 into 10 which is equal to 15 + 100 which equals 115 total cost is equal to 115 rupees number line and an expression consider the expression x + 4 let us consider the variable X takes the position X on the number line since the given expression the constant 4 takes the positive value a takes the position 4 units to the right of X if the expression is X minus 4 the position a will be to the left of X consider the expression 5x plus 6 let X take the position X on the number line since the first term of the given expression is 5x the position of 5x will be point D the distance of G from the origin will be five times the distance of X from the origin therefore the position of e of 5x plus 6 will be 6 units to the right of D algebraic expressions constants a symbol having a fixed numerical value is called a constant example 3 minus 5 9 upon 7 P etc variable a symbol or alphabet which takes on various numerical values is known as variable example circumference of a circle C is equal to 2 P R where R is radius of the circle to be constants C R Able's involved have only non-negative integral powers is called a polynomial examples 1 3 – 4 X + 9 X square plus 5 by 4 X cube is a polynomial in one variable X – 5 + 4 x squared – 7 X square y plus 9 Y minus 1 upon 3 X square square is a polynomial in two variables x and y 3 3 + 9 X raised to the power 5 upon 2 + 5 X square is not a polynomial 5 upon 2 is not a non-negative integer 4 7 + 2 X raised to the power minus 8 + 6 X square is not a polynomial – 8 is not a non-negative integer degree of a polynomial the highest power of any term in the polynomial is called degree of a polynomial s it is not a non-negative integer degree of a polynomial the highest power of any term in the polynomial is called degree of a polynomial based on the degree the polynomial are classified as linear polynomial quadratic polynomial cubic polynomial by quadratic polynomial etc linear polynomial a polynomial of degree 1 is called a linear polynomial example 3 plus 7.5 X here X is of degree 1 therefore it is a linear polynomial quadratic polynomial a polynomial of degree 2 is called a quadratic polynomial example X square minus 4x plus 4 here X of degree 2 therefore it is called as quadratic polynomial cubic polynomial a polynomial of degree 3 is called a cubic polynomial example X cubed minus 3x squared plus 5x plus 1 here X of degree 3 therefore it is a cubic polynomial by quadratic polynomial a polynomial of degree 4 is called a by quadratic polynomial example 2 minus 5x + X square minus 8x raised to the power 4 here X is of degree 4 then it is by quadratic polynomial degree of a polynomial in two or more variables in the case of polynomials in more than one variable the sum of the parts of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial example 5 x squared Y cube minus 3 XY squared plus 6 minus square root 2 X raised to the power 4 is a polynomial of degree 5 in X&Y monomials binomials and polynomials based on the number of terms of the polynomial they are classified as monomial binomial trinomial and so on monomial a polynomial is said to be a monomial if it contains one term Harry's having seven roses then it is denoted by 7x here 7x is a monomial binomial a polynomial is said to be a binomial if it contains two terms example harry's having nine roses and five jellies then we form an algebraic expression as 9x plus 5y X denotes number of roses y denotes the number of jellies 9x plus 5y is a polynomial which contains two terms trinomial a polynomial is said to be a trinomial if it contains three terms for example hair is having eight roses six jellies and five chocolate then the algebraic expression is given by 8x + 6 y + 5 Zent here x denotes number of roses y denotes number of jellies Z denotes number of chocolates 8 x + 6 y + 5 Z is a trinomial which contains three terms polynomial in general if the expression contains one or more terms whose coefficient is non zero then it is called a polynomial example 4x + 4 XY + 17 X Z square minus 10y squared Z plus 19 multiplying a binomial by a trinomial in this multiplication we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial a plus four into a square plus 5a plus seven is equal to a into a square plus 5a plus seven plus 4 into a square plus 5a plus seven which is equal to a cube plus five a square plus 7a plus 4 a squared plus 20 a plus 28 which equals AQ + 9 a square plus 27 a plus 28 example simplify a plus C into 2a minus 3b plus C minus 5 a minus 2 B into C solution a plus C into 2a minus 3b plus C is equal to a into 2a – 3 B plus C plus C into 2a minus 3b plus C is equal to 2 a square minus 3 a B plus AC plus 2 AC – 3 B C plus C square which equals 2a squared minus 3 a B + 3 AC – 3 B C plus C square 5 a – 2 B into C is equal to 5 AC minus 2 BC therefore a plus C into 2a – 3 B plus C + 5 a minus 2 B into C is equal to 2a squared + 8 AC – 5 BC minus 3 a B plus C square identity what is an identity consider the equality a plus 5 into a plus 3 is equal to a square plus 8 a plus 15 we shall evaluate both sides of this equality for some value of a say a is equal to 10 for a is equal to 10 LHS is equal to a plus 5 into a plus 3 is equal to 10 plus 5 into 10 plus 3 which equals 15 into 13 which is equal to 195 RHS is equal to a square plus eight a plus 15 which equals ten square plus 18 plus 15 is equal to 195 the values of the two sides of the Equality are equal for a is equal to 10 let us now take a as minus 1 LHS is equal to minus 1 plus 5 into minus 1 plus 3 is equal to 4 into 2 is equal to 8 RH s is equal to minus 1 square plus 8 into minus 1 plus 15 is equal to 1 minus 8 plus 15 is equal to 16 minus 8 is equal to 8 4 is equal to minus 1 LHS is equal to RHS we shall find that for any value of a LHS is equal to RHS such an equality which is true for every value of the variable in it is called an identity thus a plus 5 into a plus 3 is equal to a square plus 8 a plus 15 is an identity equation an equation is true for only certain values of the variables in it it is not true for all values of the variable a square plus 8 a plus 15 is equal to 195 is true for a is equal to 10 but it is not true for a is equal to -1 or a is equal to 0 etc Samri let us summarize what we have learned a combination of constants and variables connected by plus minus X and y is known as algebraic expression a symbol having a fixed numerical value is called a constant a symbol which takes on various numerical value is known as variable terms are added to form expressions expression contain exactly 1 2 & 3 terms are called monomial binomial and trinomial respectively an algebraic expression in which the variables involved have only non-negative integral parts is called a polynomial like terms one formed from the same variables and the parts of these variables are the same coefficients of like terms need not be the same while adding polynomial arranged the like terms unlike terms separately and add the like terms only the same rule is followed for subtraction also for multiplication each term of one polynomial is multiplied by each term of the other polynomial you

2 Comments

  1. rekha adhiya said:

    cost of 3 chocolates=3y and cost of 10 pens=10x (u hv written wrong)

    June 29, 2019
    Reply
  2. Anuj Yadav said:

    good

    June 29, 2019
    Reply

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