CLASS-6
DIVISION OF ALGEBRAIC EXPRESSION
DIVISION OF ALGEBRAIC EXPRESSION -
Division of algebraic expressions is a fundamental operation in algebra, just like addition, subtraction, and multiplication. It involves dividing one algebraic expression (usually a polynomial) by another algebraic expression. The result of this division is often referred to as the quotient.
Here are the key rules and definitions for division of algebraic expressions:
The algebraic definition of division
a 1
------- = a . -------
b b
Division, in algebra, is defined as multiplication by the reciprocal.
Rule:-
For dividing a
polynomial by a monomial, divide each
term of the polynomial by the monomial. We
divide each term of the polynomial by the monomial and then simplify.
Other way Of understanding -
Definition:-
When you divide one algebraic expression A(x) by another algebraic expression B(x), the result is a third algebraic expression Q(x) such that A(x) = B(x) ⋅ Q(x) + R(x), where R(x) is the remainder, and the degree of R(x) is less than the degree of B(x).
Rules:-
- Long Division Method:- One common method for dividing algebraic expressions is long division, which is similar to long division with numbers.Divide the term with the highest degree in the numerator (dividend) by the term with the highest degree in the denominator (divisor). Write down the result as the leading term in the quotient. Multiply the divisor by the result from the previous step and subtract it from the dividend. Bring down the next term from the dividend and repeat the process until all terms have been considered. The remainder, if any, will have a degree lower than the divisor.
- Synthetic Division Method:- This method is primarily used when the divisor is a linear expression of the form (x − c), where c is a constant. Write down the coefficients of the terms in the dividend. Set up a synthetic division table with the constant c as the root. Perform synthetic division, starting with the first coefficient. Continue until all coefficients have been considered.The result is the quotient, and there may or may not be a remainder.
- Polynomial Division Rules:- The degree of the quotient is equal to the degree of the dividend minus the degree of the divisor.If the degree of the dividend is less than the degree of the divisor, the quotient is zero.If there is a remainder (i.e., the degree of the remainder is greater than or equal to zero), it is written as R(x), and the division is expressed as A(x) = B(x) ⋅ Q(x) + R(x). If the remainder is zero, it means that the divisor is a factor of the dividend, and the division is exact.
These rules and methods are essential for solving algebraic equations and simplifying expressions involving division, especially when dealing with polynomial expressions. Division allows us to break down complex algebraic expressions and find solutions to equations involving those expressions.