Quadratic Polynomial — Definition, Formula & Examples
Key Formula
ax2+bx+c
Where:
- a = Coefficient of the squared term (must not be 0)
- b = Coefficient of the linear term
- c = 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 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
- Polynomial — General family that includes quadratics
- Degree of a Polynomial — Quadratics are defined by having degree 2
- Quadratic Equation — Setting a quadratic polynomial equal to zero
- Linear Polynomial — Degree-1 polynomial, one step below quadratic
- Cubic Polynomial — Degree-3 polynomial, one step above quadratic