1. Definition of algebraic expression:
An algebraic expression is a mathematical term that combines constants and variables using operations like addition, subtraction, multiplication, division.
Constants:
A constant is a fixed numerical value.
Variable:
A symbol which takes on various numerical values is knowns as variable or literal.
Example:
\(a + b\) , \(x^2 + x + 5\)
2. Definition of Polynomial:
An expression of the form \(p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \(n\) is an non negative integer
and \(a_0, a_1, a_2, ..., a_n\) are constants and \(a_n \neq 0\) is called a polynomial in \(x\) of degree \(n\).
Example:
\(p(x) = x^7 - 3x^6 + x^3 +x + 9\)
3. Degree of a polynomial:
If \(p(x)\) is a polynomial in \(x\), then the heightest power of \(x\) in \(p(x)\) is called degree of a polynomial \(p(x)\).
Example:
Degree of the given polynomial \(p(x) = x^7 - 3x^6 + x^3 +x + 9\) is \(7\).
4. Terms and Coefficients of Polynomials
In a polynomial, \(\text{terms}\) are the individual parts separated by plus (+) or minus (-) signs, which can contain numbers, variables, or their products.
\(\text{Coefficients}\) refer to the numerical values that are multiplied by the variable.
Example:
\(3x+5\)
Here the terms are \(3x\) and \(5\), and the coefficient of \(x\) is \(3\).