ALGEBRAIC EXPRESSIONS

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 expression 2x-3y.

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 – a are trinomials.

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 terms2. 3xy, -5ya = Unlike terms3. 6 abc, -5acd = Unlike terms4. 8pq, -3qp = Like terms

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

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 of two or more monomials is equal to the of the variables in the given monomials.

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

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

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 .

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 + 62. (a + b) = a + b + 2ab For e.g. – (3p + 4q) (3p + 4q) = (3p)² + (4q)² + 2 x 3p x 4q = 9p² + (16q)² + 24pq.

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.

PROPERTIES OF D

IVIS

IONOF RATIONAL NUMBERS

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.

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 x²2. 2 X 2 X x²3. 2 X 2 X x X x4. 4 X x X x 1 is a factor of every algebraic term, so 1 is

called a trivial factor.

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) (3y + 1)

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Made By MANAV Gupta

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