**Algebraic Expressions And Algebraic Formulas:**

The following topics have been discussed in Chapter – 4.

- Algebraic Expressions
- Algebraic Formula
- Surds and their Application
- Rationalization
**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 X^{3 }+ 5 Y^{2},

5 a + 3bc +8b

4p^{2 }– 9q^{2} + 3pq

https://www.youtube.com/watch?v=ktG5vGni9WU

https://www.youtube.com/watch?v=72CWwFK9u2w

**Polynomials:**

An Agebraic Expression of the form

P (x) = a_{n }x^{n} +a_{n-1} x^{n-1} + a_{n-2 }x^{n-2} + . . . . + a_{1} x + a_{0} where a_{n} 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 a_{n} of the highest power of x is called the leading coefficient.

The Algebraic Expression: 2 x^{4} y^{3} + x^{2}y^{2} + 8x is a Polynomial of two variables x and y and has degree 7.

Rational Expressions Behave like Rational Numbers

**Rational Expression:**

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.