Algebraic identities
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The document provides an introduction to algebraic expressions, explaining fundamental concepts such as variables, terms, and the classification of expressions such as monomials, binomials, and trinomials. It details operations involving algebraic expressions, including multiplication and factorization, as well as rules for signs in division. Additionally, it offers historical context about the development of algebra and notable mathematicians who contributed to its evolution.
Algebraic identities
- 2. INTRODUCTION • The rules in mathematics are called algebra. Algebra is the grammar of math. The rules of algebra tell how this equation can be formed and changed. Algebra is a way of thinking about arithmetic in a general way. Instead of using specific numbers, mathematicians have found that the easiest way to write down the general rules of algebra is to use what are called variables. Variables are symbols (usually letters like a, b, x, y) that represent either any number or an unknown number. For example, one of the rules of algebra says that a + b = b + a, where a and b are variables that represent any numbers.
- 3. What is Algebraic Expression? • A number or a combination of numbers connected by the symbols of operation +,-,*,/ is called an algebraic expression. E.g.- 3x, 2x, -3/5x . The no’s 3, 2, - 3, -3/5 used above are constants and the literal no’s x, y, z are variables. The several parts are called terms. The signs + and – connect the different terms. 2x and -3y are terms of the
- 4. What are Monomial, Binomial and Trinomial? An expression having only one term is called monomial. E.g.- 3x, -4y, 2/3yz are monomials. An expression having two terns is called binomial. E.g.- 2x -3y, 4x + 5xz are binomials. An expression having three terms is called trinomial. E.g.- -3p + 5q -2/5pq, 7a - 2abc –
- 5. What are Like and Unlike terms? • Terms having same combinations of literal numbers are called like terms. • Terms do not having same combinations of literal numbers are called unlike terms. For e.g.- 1. 4ab, -3ba = Like terms 2. 3xy, -5ya = Unlike terms 3. 6 abc, -5acd = Unlike terms 4. 8pq, -3qp = Like terms
- 6. What is Multiplication of Algebraic Expressions? • The product of two numbers of like signs is positive and the product of two numbers of unlike signs is negative. For E.g., - 3 x 5 = 15; -4/5 x -5/3 = 4/3 -4 x 2 = -8; 3 x (-4/3) = -4 We also know the following laws of exponents • a x a = a • (a ) = a
- 7. What is Multiplication of a Monomial by a Monomial? • The product of two monomials is obtained by the application of the laws of exponents and the rules of signs, e.g., • 2x y³ X 3x² y³ = (2x3) x X y = 6x y • Thus we have the following rules- 1. The numerical coefficient of the product of two or more monomials is equal to the product of their numerical coefficients. 2. The variable part of the product
- 8. What is Multiplication of a Binomial by a Monomial? • To multiply a binomial by a monomial, we use the following rule- a x (b + c) = a x b + a x c For E.g.- Multiply: 4b + 6 by 3a Product = 3a(4b + 6) = 3a x 4b + 3a x 6 = 12ab + 18a
- 9. What is Multiplication of a Trinomial by a Monomial? To multiply a trinomial by a monomial, we use the following rule: a x (b + c + d) = a x b + a x c + a x d For e.g. – Multiply: 3x – 2x + 2 by 3x Product = 3x(3x – 2x + 2) = 3x x 3x – 3x x 2x + 3x x 2 = 9x² - 6x² + 6x
- 10. What is Multiplication of a Polynomial by a Polynomial? • Let us multiply two binomials (4x – y) and (3x – 2y). Here we will use the law of multiplication of a binomial by a monomial twice. Consider (4x – y) as one number. Then (4x – y) (3x – 2y) = (4x –y) x 3 + (4x –y) x (-2y) = 4x x 3x – y x 3x -4x x 2y + y x 2y =12x² - 3xy – 8xy + 2y² = 12x² - 11xy + 2y² .
- 11. Which are special Products (Identities)? The four special identities are: 1. (x + a) (x + b) = x + (a + b)x + ab For e.g. – (x + 3) (x + 2) = x + (3 + 2)x + 3 x 2 = x + 5x + 6 2. (a + b) = a + b + 2ab For e.g. – (3p + 4q) (3p + 4q) = (3p)² + (4q)² + 2 x 3p x 4q = 9p² + (16q)² +
- 12. 3. (a – b) = a + b - 2ab For e.g.- (3p – 4q) (3p – 4q) = (3p)² + (4q)² -2 x 3p x 4q = (9p)² + (16q)² - 24pq. 4. (a - b ) = (a + b) (a – b) For e.g.- (a – b) - (a + b) = (a – b + a + b) (a – b – a – b) =2a x (-2b) =-4ab.
- 13. What are rules of signs in Division? 1. When the dividend and the divisor have the same signs, the quotient has the plus sign. 2. When the dividend and the divisor have opposite signs, the quotient has the negative sign. 3. The process of division may be divided in three cases: • Division of a monomial by another monomial. • Division of a polynomial by monomial. • Division of polynomial by another polynomial.
- 14. What is Factorization of Algebraic Expressions? The factors of- • a+ 2ab + b are (a+b) (a+b) • A – 2ab + b are (a-b) (a-b) • A -b are (a-b) (a +b) • 4x are- 1. 4 X x² 2. 2 X 2 X x² 3. 2 X 2 X x X x 4. 4 X x X x • 1 is a factor of every algebraic term, so 1 is called a trivial factor.
- 15. How do we do Factorization by Regrouping Terms? • Sometimes it is not possible to find the greatest common factor of the given set of monomials. But by regrouping the given terms, we can find the factors of the given expression. For e.g.,- 3xy + 2 + 6y + x = 3xy + 6y + x + 2 = 3y(x + 2) + 1(x + 2) =(x + 2)
- 16. Some Other Information- • The word algebra comes from the title of a book on mathematics written in the early 800s by an Arab astronomer and mathematician named al-Khwarizmi. The rules of algebra are older than that, however. The ancient Greeks wrote down some of the rules that make up algebra, but others came later. In the 500s Hindu mathematicians in India added the idea of 0. One of the final steps in the development of modern algebra came in the 1600s, when mathematicians developed the idea of negative numbers. Although the ancient Chinese and others had a way to indicate negative numbers, it was not until the 1600s that they were properly understood.
- 17. Made by : Samyak Jain- 04