Maths Teaching Guide: Algebraic Expressions

Maths Teaching Guide Algebraic Expressions 6 Algebraic ExpressionsYou knowto write the foothold, coefficients and factors of an algebraicalalal feel.to classify an algebraic expression as monomial, binomial, trinomial.to identify like end doses.to add and subtract algebraic expression.You get out learnmultiplication and division of given multinomials.the difference between an personal individualism and an equating.algebraic identities and their applications.factorization of algebraic expression by regrouping , by winning common factors or development algebraic identities.Let us regain the basic definitions of algebraConstants and variables A quantity having a fixed numerical protect is called a constant whereas variables in algebra atomic number 18 letters much(prenominal)(prenominal) as x, y, z or any some other letter that muckle be used to represent unknown numbers.Algebraic expression An expression which has a combination of constants and variables connected t o each other by one or more operation (+,-,X,) is called an algebraic expression. vitrine be all algebraic expressionsTerm The parts of an algebraic expression separated by an amplification or a subtraction concentrate be called cost of the expression. In the expression the endpoints of the expression are are variable shapes as their values exit form with the value of x, while (-4) is a constant term.On the basis of the number of terms in an algebraic expression, they are classified as monomials, binomials, trinomials and polynomials.Monomials are algebraic expressions having one term .Binomials are algebraic expressions having dickens terms.Trinomials are algebraic expressions having three terms.Polynomials are algebraic expressions having one or more than one term.Remember unaccompanied expressions with positive powers of variables are called polynomials. An expression of the type is not a polynomial as and the power of variable p is (- 1) which is not a all to ld number. casing 1Classify the algebraic expressions as monomials, binomials or trinomials. declarationbinomialmonomialtrinomialmonomialbinomialLike and contrary terms Terms having the uniform algebraic factors are called like terms . The numerical coefficients may be different. 2x2yz, 5x2yz, 8x2yz and 2x2yz are like terms3p 3q2, 7p 3q2and 9p 3q2 are withal like terms.Unlike terms Terms having different algebraic factors are called unlike terms, , 3x2yz3p 3q2 are unlike terms. numberition and Subtraction of Algebraic Expressions.In algebra, like terms can be added or subtracted.To add or subtract algebraic expressions we can use the flat method or the newspaper tug method.The swimming methodAll algebraic expressions are written in a horizontal line the like terms are wherefore grouped. The congeries or difference of the numerical coefficients is thusly found. case 2 augment the succeeding(a) stemExample 3Subtract SolutionThe column methodIn the column method, each expr ession is written in a separate row in such a fashion that like terms are arranged one below the other in a column. The sum or difference of the numerical coefficients is then found.Example 4Add SolutionTo add by horizontal method, collect the like terms and add coefficients.To add by column method, arrange the like terms in column and addExample 5Subtract SolutionWe know that the subtraction of two algebraic expressions or terms is addition of the additive inverse of the second term to the graduation exercise term. Since the additive inverse of a term has opposite sign of the term, hence we can read that in subtraction of algebraic expressions change + to and change to + for the term to be subtracted and then add the two termsTo subtract by column method, arrange the like terms in columns and change the sign of the subtrahendExample 6What should be added to to get SolutionThe expression to be added will beExercise 6.1Classify the algebraic expressions as monomials, binomial s or trinomials. Also write the terms of the expressionAdd the following algebraic expressions by the horizontal methodAdd the following algebraic expressions.Subtract the following expressions.Subtract the sum of from the sum of . twain adjacent sides of a rectangle are . What will be the gross profit of the rectangle.The perimeter of a triangle is and the throwaway of two sides is. What will be the measure of the third side?What should be added to to get .What should be subtracted from to getBy how much is greater than .Multiplication of Algebraic ExpressionsMultiplication of a monomial by another monomialTo reckon 2 monomials compute the numerical coefficients compute the existent coefficients and use jurisprudences of exponents if variables are same.The everywherelap of two monomials is always a monomial.Example 1 watch over the harvest-feast ofSolutionGeometrical recital of harvest-feast of two monomialsThe knowledge domain of a rectangle is given by the product of space and width.If we moot the length as l and pretentiousness as b, thenArea of rectangle = l x bThus, it can be express that the theater of operations of a rectangle is product of two monomials.Let us consider a rectangle of length 4p and fullness 3p,Area of rectangle ABCD =AB x AD = 4p x 3p = 12p2Multiplication of a monomial by a binomialTo multiply a monomial by a binomial, we use the distributive law regurgitate the monomial by the first term reproduce the monomial by the second term of the binomial.The result is the sum of the two termsThe product of a monomial and a binomial is always a binomial.Example 2Find the productSolutionExample 3Multiply SolutionGeometrical interpretation of product of a monomial and a binomialArea of rectangle = l x bLet us draw a rectangle ABCD with length (p+q) and breadth k.Take a point P on AB such that AP = p and PB = q. magnet a line twin to AD from the point P, PQAD coming together DC at Q.Area of rectangle ABCD = sports stadium o f rectangle APQD +area of rectangle PBCQ= k x p + k x q= k(p + q)Thus, the product k(p + q) represents the area of a rectangle with length as a binomial (p+q) and breadth as a monomial k.Multiplication of a monomial by a polynomialTo multiply a monomial with a binomial, we can extend the distributive law furtherThe product of a monomial and a polynomial is a polynomial.Example 3Find the product of SolutionWe micturate multiplied horizontally in all the above examplesWe can also multiply vertically as shown belowMultiply Geometrical interpretation of product of a monomial and a polynomialLet us consider a rectangle with length = (p +q + r) and breadth= kTake points M and N on AB such thatAM = p and MN = q and NB = r.from the points M and N draw parallel to AD,MXAD and NYAD meeting DC at X and Y.Area of rectangle ABCD = area of rectangle AMXD +area of rectangle MNYX +area of rectangle NBCYArea of rectangle ABCD=pk + qk + rk = k(p + q+ r)Thus, the product of a monomial and a polynomi al represents the area of a reactangle with length as a polynomial and breadth as a monomial.Example 4Simplify SolutionMultiplication of binomialsTo multiply two binomials (a + b) and (c + d) we will again use the distributive law of multiplication over addition twiceExample 5Multiply SolutionWe have multiplied horizontally in all the above examplesWe can also multiply vertically as shown belowMultiplication of polynomial by a polynomialA polynomial is an algebraic expression having 1 or more than one termTo multiply two polynomials, we will use the distributive property that is multiply each term of the first polynomial with each term of the second polynomial.Example 6Multiply SolutionWe have multiplied horizontally in the above example, We can also multiply vertically as shown belowExercise 6.2Multiply the following monomials2a and 9bFind the following products and evaluate for x = 1, y = -1Find the following products by horizontal methodFind the following products using column me thodFind the area of the rectangle with the given measurementsLength = 3p, breadth = 4pLength = (2a+4), breadth = 5aMultiply the followingSimplify the following expressionsMultiply .Simplify If the length of a rectangle is and breadth is 3abc,find the area of the rectangle.Algebraic identitiesAn identity is a special type of equation in which the LHS and the RHS are touch on for all values of the variables.The above equation is true for all practical values of a and b so it is called an identity.An identity is different from equation as an equation is not true for all values of variables,it has a unique solution.Example There are a number of identities which are used in mathematics to make calculations easy. We are going to write up 4 basic identitiesVerification of identitiesin this identity a and b can be positive or negativeGeometrical bridle of identitiesGeometrical demonstration for.Draw a unbowed with length as shown in the figure.Let the area of reliable square be Xth en, area of full-strength PQRS=(side)2 ,Mark a point M on PQ such that length of PM = a and length of MQ= b.Draw a line MC parallel to PS intersecting SR at C.Similarly, mark a point B on RQ such that RB = a and QB = b.Draw a line BD parallel to QP intersecting PS at D.The whole square is divided into 2 squares and 2 rectangles say A1, A4,A2and A3Area of Square X1 = side2= a2Area of rectangle X2= length x breadth = abArea of rectangle X3= length x breadth = abArea of Square X4 = side2= b2area of Square PQRS = sum of inside area = area of X1+ area of X2+ area ofX3+ area ofX4Geometrically demonstration for .We draw a square with length a as shown in the figure.Let the area of original square is AThen, area of Square PQRS=(side)2 Mark a point M on PQ such that the length of PM = a-b and length of MQ= b.Draw a line MC parallel to PS intersecting SR at C.Similarly, mark a point B on RQ such that RB = a b and QB = b.Draw a line BD parallel to QP intersecting PS at D.The whole square