Polynomials – Class 9 Concepts & Practice
Polynomials – Class 9 Concepts & Practice
Concept 1 – What is a Polynomial?
What is a Polynomial?
\( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)
All exponents must be non-negative whole numbers; coefficients are real numbers
A polynomial in one variable \( x \) is an algebraic expression where every term has the form \( a \cdot x^k \) with \( k \) a whole number (0, 1, 2, …) and \( a \) a real number. Expressions involving \( \frac{1}{x} \), \( \sqrt{x} \), or negative powers of \( x \) are NOT polynomials.
📖 Examples and Non-Examples
\( 3x^2 - 5x + 2 \) → Polynomial ✓
\( x^3 + \sqrt{2}\, x - 1 \) → Polynomial ✓ (\( \sqrt{2} \) is a valid real coefficient)
\( \frac{1}{x} + 2 = x^{-1} + 2 \) → NOT a polynomial ✗ (negative exponent)
\( \sqrt{x} + 1 = x^{1/2} + 1 \) → NOT a polynomial ✗ (fractional exponent)
💡 Key rule: All powers of the variable must be non-negative whole numbers. Coefficients can be any real number including surds like \( \sqrt{3} \).