ALGEBRAIC EXPRESSIONS AND IDENTITIES

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ALGEBRAIC EXPRESSIONS AND IDENTITIES : 

ALGEBRAIC EXPRESSIONS AND IDENTITIES MADE BY – PRATEEK SINHA CLASS – VIII-C ROLL NO. - 41

WHAT ARE ALGEBRAIC EXPRESSIONS? : 

WHAT ARE ALGEBRAIC EXPRESSIONS?

Slide 3: 

A NUMBER OR A COMBINATION OF NUMBERS CONNECTED BY THE SYMBOLS OF OPERATION +, -, x,  IS CALLED AN ALGEBRAIC EXPRESSION. EQUATIONS ARE FORMED FROM VARIABLES AND CONSTANTS.

TERMS AND FACTORS : 

TERMS AND FACTORS VARIOUS PARTS OF AN ALGEBRAIC EXPRESSION WHICH ARE SEPERATED BY THE SIGNS + OR – ARE CALLED THE ”TERMS” OF THE EXPRESSION. EACH TERM IN AN ALGEBRAIC EXPRESSION IS A PRODUCT OF ONE OR MORE NUMBERS KNOWN AS THE “FACTORS” OF THAT TERM. FOR EX : FACTORS OF 24 =

LIKE AND UNLIKE TERMS : 

LIKE AND UNLIKE TERMS THE TERMS HAVING THE SAME LITERAL FACTORS ARE CALLED LIKE TERMS, OTHERWISE THEY ARE CALLED UNLIKE TERMS.

COEFFICIENT : 

COEFFICIENT IN A TERM OF AN ALGEBRAIC EXPRESSION ANY OF THE FACTORS WITH THE SIGN OF THE TERM IS CALLED THE COEFFICIENT OF THE PRODUCT OF OTHER FACTORS.

MONOMIALS, BINOMIALS, TRINOMIALS AND POLYNOMIALS : 

MONOMIALS, BINOMIALS, TRINOMIALS AND POLYNOMIALS EXPRESSION THAT CONTAINS ONLY ONE TER IS CALLED MONOMIAL. EXPRESSION THAT CONTAINS TWO TERMS IS CALLED BINOMIAL. EXPRESSION CONTAINING THREE TERMS IS CALLED TRINOMIALS EXPRESSION CONTAINING ONE OR MORE TERMS IS CALLED POLYNOMIALS.

ADDITION OF ALGEBRAIC EXPRESSIONS : 

ADDITION OF ALGEBRAIC EXPRESSIONS EX. ADD – (3x3 + 2x – 5) + (2x3 + x + 9) = 3x3 + 2x3 + 2x + x -5 + 9 {Collecting Like Terms} = (3 + 2)x3 + (2 + 1)x + (-5 + 9) {Adding Like Terms} = 5x3 + 3x + 4 ……………….THIS IS HOW WE ADD ALGEBRAIC EXPRESSIONS.

SUBTRACTION OF ALGEBRAIC EXPRESSIONS : 

SUBTRACTION OF ALGEBRAIC EXPRESSIONS EX. SUBTRACT – (4x3 + x2 + x + 6) – (2x3 - 4x2 + 3x + 5) = 4x3 + x2 + x + 6 - 2x3 + 4x2 - 3x – 5 = 4x3 - 2x3 + x2 + 4x2 + x -3x + 6 – 5 = 2x3 + 5x2 – 2x + 1 ……………….THIS IS HOW WE SUBTRACT ALGEBRAIC EXPRESSIONS.

MULTIPLICATION OF TWO MONOMIALS : 

MULTIPLICATION OF TWO MONOMIALS EX. MULTIPLY : (3ab) x (5b) = (3 x 5) x (ab x b) = 3 x 5 x [a x (b x b)] = 15 x [a x b2] = 15 x ab2 = 15ab2 ……………….THIS IS HOW WE MULTIPLY TWO MONOMIALS.

MULTIPLICATION OF A MONOMIAL AND A BINOMIAL : 

MULTIPLICATION OF A MONOMIAL AND A BINOMIAL EX. MULTIPLY : 2x by (3x + 5y) = (2x x 3x) + (2x x 5y) = 6x2 + 10xy ……………….THIS IS HOW WE MULTIPLY A MONOMIAL BY A BINOMIAL.

MULTIPLICATION OF A MINOMIAL BY A TRINOMIAL : 

MULTIPLICATION OF A MINOMIAL BY A TRINOMIAL EX. MULTIPLY : 3p x (4p2 + 5p + 7) = (3p x 3p2) + (3p x 5p) + (3p x 7) = 9p3 + 15p2 + 21p ……………….THIS IS HOW WE MULTIPLY A MONOMIAL BY A TRINOMIAL.

MULTIPLICATION OF TWO BINOMIALS : 

MULTIPLICATION OF TWO BINOMIALS EX: MULTIPLY – (3x + 2y) and (5x + 3y) = (3x + 2y) x (5x + 3y) = 3x x (5x + 3y) + 2y x (5x + 3y) = (3x x 5x + 3x x 3y) + (2y x 5x + 2y x 3y) = 15x2 + 9xy + 10xy + 6y2 = 15x2 + 19xy + 6y2 ……………….THIS IS HOW WE MULTIPLY A MONOMIAL BY A TRINOMIAL.

MULTIPLICATION OF A BINOMIAL BY A TRINOMIAL : 

MULTIPLICATION OF A BINOMIAL BY A TRINOMIAL EX: MULTIPLY – (a + b) (2a – 3b + c) - (2a - 3b + c) = a (2a – 3b + c) + b (2a – 3b + c) = 2a2 – 3ab + ac + 2ab – 3b2 + bc = 2a2 – ab + ac – 3b2 + bc ……………….THIS IS HOW WE MULTIPLY A MONOMIAL BY A TRINOMIAL.

WHAT IS AN IDENTITY? : 

WHAT IS AN IDENTITY?

Slide 16: 

AN IDENTITY IS AN EQUALITY WHICH IS TRUE FOR ALL VALUES OF THE VARIABLE.

STANDARD IDENTITIES : 

STANDARD IDENTITIES

Slide 18: 

IDENTITIY 1 (a + b)2 = a2 + 2ab + b2 or, (a + b)2 = a2 + b2 + 2ab i.e., Square of the sum of two terms = (Square of the first term) + (Square of the second term) + 2 x (First term) x (Second term) Proof: (a + b)2 = (a + b)(a + b) =(a + b)2 = a (a + b) + b (a + b) =(a + b)2 = a2 + ab + ba + b2 =(a + b)2 = a2 + ab + ab + b2 =(a + b)2 = a2 + 2ab + b2 =(a + b)2 = a2 + b2 + 2ab

Slide 19: 

IDENTITIY 2 (a - b)2 = a2 - 2ab + b2 or, (a - b)2 = a2 + b2 - 2ab i.e., Square of the difference of two terms = (Square of the first term) + (Square of the second term) + 2 x (First term) x (Second term) Proof: (a - b)2 = (a - b)(a - b) =(a - b)2 = a (a - b) - b (a - b) =(a - b)2 = a2 - ab - ba + b2 =(a - b)2 = a2 - ab - ab + b2 =(a - b)2 = a2 - 2ab + b2 =(a - b)2 = a2 + b2 - 2ab

Slide 20: 

IDENTITIY 3 (a + b)(a - c) = a2 – b2 i.e., (First term + Second term) (First term - Second term) = (First term)2 – (Second term)2 Proof: (a + b)(a - b) = a (a - b) + b (a - b) =(a + b)(a - b) = a2 - ab + ba – b2 =(a + b)(a – b) = a2 – ab + ab – b2 =(a + b)(a – b) = a2 – b2

THANK YOU : 

THANK YOU