Chapter 2: Polynomials

What is a Polynomial?

Definition

A polynomial in variable xx is an expression of the form:

p(x)=anxn+an−1xn−1+...+a1x+a0p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0

Where:

  • Each aia_i is a coefficient (a constant number)
  • Each power of xx is a non-negative integer (0, 1, 2, 3...)
  • an≠0a_n \neq 0 (leading coefficient is not zero)

Examples vs Non-Examples

✅ These ARE Polynomials:

ExpressionWhy?
3x2+2x+13x^2 + 2x + 1All powers are whole numbers
5y3−4y+75y^3 - 4y + 7Valid - uses variable y
77Constant polynomial (degree 0)
xxLinear polynomial (degree 1)

❌ These are NOT Polynomials:

ExpressionWhy NOT?
x−1+2x^{-1} + 2Negative exponent
x+1\sqrt{x} + 1Fractional exponent (x1/2x^{1/2})
1x\frac{1}{x}Same as x−1x^{-1}
x+1x−1\frac{x+1}{x-1}Variable in denominator

Key Terms

TermMeaningExample
CoefficientNumber in front of variableIn 5x25x^2, coefficient is 5
DegreeHighest power of variableDegree of x3+2xx^3 + 2x is 3
Constant termTerm without variableIn x2+3x+7x^2 + 3x + 7, it is 7
Visualizer
2x³ - 5x² + 3x - 7
Degree: 3 (highest power)
2x³
Coefficient: 2
Power: x^3
(cubic term)
-5x²
Coefficient: -5
Power: x^2
(quadratic term)
+3x
Coefficient: 3
Power: x^1
(linear term)
-7
Coefficient: -7
Power: x^0
(constant term)

Interactive Visualization