Polynomials – Concepts & Practice
Polynomials – Concepts & Practice
Concept 1 – Introduction to Polynomials
Introduction to Polynomials
\(p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where \(a_n \neq 0\)
Each \(a_k x^k\) is called a term; \(a_k\) are coefficients; exponents must be whole numbers
A polynomial in one variable \(x\) is an algebraic expression of the form \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where each exponent is a non-negative integer and each coefficient is a real number. Expressions with negative exponents, fractional exponents, or variables in the denominator are NOT polynomials. For example, \(x^{-1}\), \(\sqrt{x} = x^{\frac{1}{2}}\), and \(\frac{1}{x}\) are not polynomials.
📖 Polynomial vs Non-Polynomial
\(3x^2 - 5x + 7\) ✓ Polynomial (whole number exponents)
\(x^3 + 4x - 2\) ✓ Polynomial
\(x^{-2} + 3x\) ✗ Not a polynomial (negative exponent)
\(\sqrt{x} + 5\) ✗ Not a polynomial (fractional exponent \(\frac{1}{2}\))
💡 Key check: All exponents of the variable must be whole numbers (0, 1, 2, 3, ...). Coefficients can be any real number, including fractions or irrationals.