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Quartic Polynomial — Definition, Formula & Examples

Quartic Polynomial

A polynomial of degree 4.

Examples: 3x4 – 2x3 + x2 + 8, a4 + 1, and m3n + m2n2 + mn.

Key Formula

f(x)=ax4+bx3+cx2+dx+ef(x) = ax^4 + bx^3 + cx^2 + dx + e
Where:
  • aa = Leading coefficient (must not be zero)
  • b,c,db, c, d = Coefficients of the cubic, quadratic, and linear terms (can be zero)
  • ee = Constant term
  • xx = Variable

Worked Example

Problem: Determine whether the polynomial 2x^4 - 7x^2 + x - 3 is a quartic polynomial, and identify its leading coefficient.
Step 1: Find the highest power of the variable.
2x4−7x2+x−3⇒highest power is 42x^4 - 7x^2 + x - 3 \quad \Rightarrow \quad \text{highest power is } 4
Step 2: Since the degree is 4, this is a quartic polynomial. The leading coefficient is the number in front of the degree-4 term.
Leading coefficient=2\text{Leading coefficient} = 2
Answer: Yes, it is a quartic polynomial with leading coefficient 2.

Why It Matters

Quartic polynomials appear in physics (e.g., elastic potential energy expressions) and in optimization problems. Unlike quintic and higher-degree polynomials, quartic equations can always be solved exactly using a formula, though the formula is far more complex than the quadratic formula. Recognizing that a polynomial is quartic tells you it can have at most four real roots and up to three turning points.

Common Mistakes

Mistake: Confusing "quartic" (degree 4) with "quadratic" (degree 2) because the words sound similar.
Correction: Remember that "quad" refers to the second power, while "quart" (like "quarter") refers to the fourth power. A quartic polynomial always has an x4x^4 term as its highest-degree term.

Related Terms