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Quadratic Polynomial — Definition, Formula & Examples

Quadratic Polynomial

A polynomial of degree 2.

Examples: x2 + 8x – 5, a2 + 7a, and mn + m2 + n2.

Key Formula

ax2+bx+cax^2 + bx + c
Where:
  • aa = Coefficient of the squared term (must not be 0)
  • bb = Coefficient of the linear term
  • cc = Constant term

Worked Example

Problem: Determine whether 3x² − 7x + 2 is a quadratic polynomial.
Step 1: Identify the term with the highest exponent. The term 3x² has exponent 2.
3x2  ⇒  degree 23x^2 \;\Rightarrow\; \text{degree } 2
Step 2: Check that no other term has a higher exponent. The remaining terms are −7x (degree 1) and 2 (degree 0). Neither exceeds 2.
Step 3: Confirm the leading coefficient is not zero. Here a = 3 ≠ 0, so the degree is genuinely 2.
Answer: Yes, 3x² − 7x + 2 is a quadratic polynomial because its highest degree is 2.

Why It Matters

Quadratic polynomials model many real-world situations, including projectile motion, area calculations, and profit optimization. Learning to recognize them is the first step toward factoring, completing the square, and using the quadratic formula to find their roots.

Common Mistakes

Mistake: Thinking a polynomial like x³ + x² is quadratic because it contains an x² term.
Correction: The degree of a polynomial is determined by its highest-degree term. Since x³ has degree 3, the polynomial is cubic, not quadratic.

Related Terms