A polynomial is an algebraic expression involving variables and coefficients, where the variables' exponents are non-negative integers (whole numbers).
General Form and Components:
The general form of a polynomial in one variable, \(x\), is:
\(p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0\)
Variable: The symbol representing the unknown value, often \(x\) (e.g., \(x, y, h, r, p\)).
Terms: The parts of the expression separated by addition or subtraction (e.g., \(a_n x^n, a_{n-1} x^{n-1}\), etc.).
A polynomial can have a finite number of terms.
Coefficients: The numerical factors multiplying the variables (e.g., \(a_n, a_{n-1}, \dots, a_0\)).
Constant Term: A term with no variable, which is a constant number (e.g., \(a_0\)).
Example:
For \(p(x) = x^2 - 8x + 9\): \(x\) is the variable, \(1\) is the coefficient of \(x^2\), \(-8\) is the coefficient of \(x\), and \(9\) is the constant.
Classification of Polynomials: Polynomials are classified based on the number of terms and the degree.
Classification by Number of Terms:
Monomial: A polynomial with one term (e.g., \(p(x) = 8x\)).
Binomial: A polynomial with two terms (e.g., \(p(x) = 9x^3 + 3\)).
Trinomial: A polynomial with three terms (e.g., \(p(x) = 7x^4 + 3x^3 + 7\)).
Classification by Degree:
The polynomial degree is the highest variable power in the polynomial.
| Degree | Name of the polynomial | General form (in \(x\) | No of terms | Example |
| \(1\) | Linear Polynomial |
\(p(x) = ax + b\)
|
\(2\) | \(p(x) = 8x - 2\) |
| \(2\) | Quadratic Polynomial | \(p(x) = ax^2 + bx + c\) | \(3\) | \(p(x) = 3x^2 + 8x - 2\) |
| \(3\) | Cubic Polynomial | \(p(x) = ax^3 + bx^2 + cx + d\) | \(4\) | \(p(x) = 9x^3 - 3x^2 + 8x - 2\) |
Important!
- Non-zero Constant Polynomial: A polynomial of the form \(p(x) = c\) (where \(c\) is a non-zero number) has a degree of zero (since \(c = c x^0\)).
- Zero Polynomial: The constant polynomial \(p(x) = 0\). Its degree is not defined.
Polynomials in More Than One Variable:
A polynomial can have multiple variables, such as two, three, or four, as long as all variable exponents are non-negative integers.
- Two variables (e.g., \(x\) and \(y\)): \(p(x, y) = xy\) (Area of a rectangle).
- Three variables (e.g., \(x, y,\) and \(h\)): \(p(x, y, h) = \frac{1}{2}(x+y) \times h\) (Area of a trapezium).
The value of the polynomial \(p(x)\) at \(x=a\) is \(p(a)\) acquired when \(x\) is replaced by \(a\) (\(a∈R\))
Zero of a Polynomial
The zero of a polynomial \(p(x)\) is a real number '\(\mathbf{a}\)' such that \(\mathbf{p(a) = 0}\).
- To find the zero, set the polynomial equal to zero: \(p(x) = 0\).
- In this situation, \(p(x) = 0\) is called a polynomial equation, and the value '\(\mathbf{a}\)' is a root of the equation.
- A non-zero constant polynomial does not have zeros.
- Every real number is a zero of the zero polynomial (\(p(x) = 0\)).