Algebraic Expressions And Algebraic Formulas:
The following topics have been discussed in Chapter – 4.
- Algebraic Expressions
- Algebraic Formula
- Surds and their Application
- Algebraic Expressions: Arithmetic is generalized in the subject of Algebra. Algebraic Terms, when subject to subtraction and/or additions, we get Algebraic Expressions. The following are some examples of Algebraic Expressions:
8 X3 + 5 Y2,
5 a + 3bc +8b
4p2 – 9q2 + 3pq
An Agebraic Expression of the form
P (x) = an xn +an-1 xn-1 + an-2 xn-2 + . . . . + a1 x + a0 where an is not equal to zero.
Degree of Polynomial: Degree of Polynomial is n when it has the highest power of x and is a non-negative integer.
Leading Coefficient of the Polynomial: The Coefficient an of the highest power of x is called the leading coefficient.
The Algebraic Expression: 2 x4 y3 + x2y2 + 8x is a Polynomial of two variables x and y and has degree 7.
Rational Expressions Behave like Rational Numbers
The quotient of two polynomials p(x) and q(x), p(x)/q(x), where q(x) is not zero, is called a Rational Expression. Every polynomial can be considered as a Rational Expression but every Rational Expression is not necessarily a Polynomial.
Rational Expression in its Lowest Form: When two Polynomials p(x) and q(x) with integral coefficients have no common factor, the rational expression p(x)/q(x), is said to be in its lowest form.
Working Rules to Reduce the Rational Expression to its Lowest Form:
- Make factors of each polynomial p(x) and q(x)
- Find H.C.F. of p(x) and q(x)
- Divide the numerator and the denominator by H.C.F.
The Rational Expression so obtained is in its lowest form.
Evaluation of Algebraic Expression at some particular Real Number:
When we substitute the variables with their specific values, the resulting number is the value of that Algebraic Expression.
Surds and their Application: An irrational radical with rational radicand is called a Surd.
Addition and Subtraction of Surds: Similar Surds, having the same irrational factors, can be added to make a single term.
Multiplication and Division of Surds: Surds of the same order can be multiplied and divided.
Rationalization of Surds:
- A Surd which contains a single term is called a monomial surd.
- A Surd which contains a sum of two monomial surds or one monomial surd and a rational number is called a binomial surd.
- When the product of the of two surds is a rational number, the each surd will be the rationalizing factor of the other.
- When a surd is multiplied by its rationalizing factor, we get a rational number. The process is called Rationalization of the given surd.