Algebraic Expressions and Identities

bySiby Sebastian PGT(Maths)

Mathematical ExpressionsMathematical Expressions

We are familiar with the following type We are familiar with the following type of of numerical expressionsnumerical expressions::

12 + 612 + 6

3 x 123 x 12

6 (3 + 2)6 (3 + 2)

15 - 4 (6)15 - 4 (6)

What is a variable?What is a variable?

Consider the expressions Consider the expressions

x + 12x + 12

3y – 23y – 2

What is a variable?What is a variable?

x + 12 , 3y – 2x + 12 , 3y – 2

These expressions involve certain These expressions involve certain letters x and y .letters x and y .

Any value can be assigned to these Any value can be assigned to these letters.letters.

A A variablevariable is a letter or symbol that is a letter or symbol that represents an unknown value.represents an unknown value.

Algebraic ExpressionsAlgebraic Expressions

When variables are used with other When variables are used with other numbers, parentheses, or operations, numbers, parentheses, or operations,

they create an they create an algebraic algebraic expressionexpression..

x + 2x + 2

(a) (b)(a) (b)

3m + 6n - 63m + 6n - 6

What is a term?What is a term?

A A termterm is the name given to a is the name given to a number, a variable, or a number, a variable, or a number and a variable number and a variable

combined by multiplication or combined by multiplication or division separated by the division separated by the operation of addition or operation of addition or

multiplication.multiplication.

TERMSTERMS

AlgebraicAlgebraic ExpressionExpression a + 2 a + 2

Terms Terms a , 2 a , 2

Algebraic ExpressionAlgebraic Expression 3m + 6n - 63m + 6n - 6

Terms 3m, 6n, - 6Terms 3m, 6n, - 6

What are coefficients?What are coefficients?

A A coefficientcoefficient is the numerical factor is the numerical factor of the term of an algebraic of the term of an algebraic

expression.expression.

Algebraic Expression TermAlgebraic Expression Term CoefficientCoefficient

6m + 56m + 5 6m 6m 66

8r + 7m + 48r + 7m + 4 7m 7m 7 7

14b - 814b - 8 14b 14b 1414

What are constants?What are constants?

A constant is a number in an A constant is a number in an expression that cannot change its expression that cannot change its value.value.

In the expression: 5x + 7y - In the expression: 5x + 7y - 22

the constant is - 2.the constant is - 2.

Find the constant in 10ab - 4b+17a - Find the constant in 10ab - 4b+17a - 50 ?50 ?

Figure it out!Figure it out!

Identify the terms, coefficients, and Identify the terms, coefficients, and constants.constants.

1. 12a - 6b + 41. 12a - 6b + 4

2. 4x - 2y2. 4x - 2y

3. c - 323. c - 32

4. 3x + 24. 3x + 2

Write algebraic expressions for Write algebraic expressions for these word phrasesthese word phrases

1.1. Four more than sFour more than s

2.2. The product of 7 and cThe product of 7 and c

3.3. Nine less than xNine less than x

4.4. A number divided by the sum of 4 A number divided by the sum of 4 and 7.and 7.

5.5. Twice the sum of a number plus 4.Twice the sum of a number plus 4.

6.6. The sum of ¾ of a number and 7.The sum of ¾ of a number and 7.

7.7. Ten times a number increased by 150.Ten times a number increased by 150.

MONOMIALSMONOMIALS

2x2x

3ab3ab

4y4y22

These expressions have only one These expressions have only one term. Such expressions are called term. Such expressions are called MONOMIALSMONOMIALS

BNOMIALSBNOMIALS

2x -3y2x -3y

3ab + 8b3ab + 8b22

4y4y22 + 16z + 16z

These are expressions having two These are expressions having two terms. Such expressions are called terms. Such expressions are called BINOMIALSBINOMIALS

TRINOMIALSTRINOMIALS

2x -3y -102x -3y -10

3ab + 8b3ab + 8b2 2 + 20 + 20

4y4y22 + 16z -5 + 16z -5

These are expressions having three These are expressions having three terms. Such expressions are called terms. Such expressions are called TRINOMIALSTRINOMIALS

POLYNOMIALSPOLYNOMIALS

Algebraic Expressions having one or Algebraic Expressions having one or more terms with non zero coefficients more terms with non zero coefficients and non negative integral powers for and non negative integral powers for the variable are called POLYNOMIALSthe variable are called POLYNOMIALS

2x , 3x+15 , 2x-3y+2,2x , 3x+15 , 2x-3y+2,

3xy – 4y + 2x3xy – 4y + 2x33 +7y +7y22 -10, -10,

2x2x55 +4xy +4xy33 +10xy-3, +10xy-3,

2ab + 4a2ab + 4a22 – 6b – 6b22 - 2a - 3b + 2 - 2a - 3b + 2

LIKE TERMS LIKE TERMS

Like terms are terms having same Like terms are terms having same variables with same powers. But variables with same powers. But their coefficients may be different.their coefficients may be different.

ExamplesExamples

1. 2xy , -5xy, 10xy,1. 2xy , -5xy, 10xy,

2. 2a2. 2a22b , -10 ab , -10 a22b, ab, a22bb

IDENTIFY LIKE TERMSIDENTIFY LIKE TERMS

3x3x33yz , 4x, -5y , yz , 4x, -5y , x x33 , 5xy, , 5xy, -5xyx-5xyx33, 2y , -, 2y , -7xyz, 10x, 27xy, 7xyz, 10x, 27xy, 22y , 6yz22y , 6yz

OPERATIONS ON OPERATIONS ON POLYNOMIALSPOLYNOMIALS

POLYNOMIALS CAN BEPOLYNOMIALS CAN BE

1.ADDED1.ADDED

2.SUBTRACTED2.SUBTRACTED

3.MULTIPLIED AND 3.MULTIPLIED AND

4.DIVIDED4.DIVIDED

QUIZQUIZ

1.1. In the expression 3x+yIn the expression 3x+y22 name the name the terms.terms.

2. What is the coefficient of -3xy? 2. What is the coefficient of -3xy?

3.3. Are 2xy and 2xAre 2xy and 2x22y like terms? y like terms?

4. Are 3x2y and -2yx2 like terms?

QUIZQUIZ

5.5. What is the maximum and What is the maximum and minimum number of terms in minimum number of terms in a polynomial?a polynomial?

6. Is 5 an algebraic expression?6. Is 5 an algebraic expression?7. What are the variables in 7. What are the variables in

the equation 7xthe equation 7x22y + 8xyy + 8xy22 ? ? Are they like terms? Are they like terms?