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Polynomial Discriminant — Definition, Formula & Examples

The polynomial discriminant is a value calculated from a polynomial's coefficients that tells you about the nature of its roots — specifically, whether the roots are real or complex, and whether any roots are repeated.

For a polynomial p(x)=anxn+an−1xn−1+⋯+a0p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0, the discriminant Δ\Delta is a polynomial expression in the coefficients that equals zero if and only if p(x)p(x) has a repeated root. For the quadratic ax2+bx+cax^2 + bx + c, the discriminant is Δ=b2−4ac\Delta = b^2 - 4ac.

Key Formula

Δ=b2−4ac\Delta = b^2 - 4ac
Where:
  • Δ\Delta = The discriminant of the quadratic
  • aa = Coefficient of $x^2$
  • bb = Coefficient of $x$
  • cc = Constant term

How It Works

Compute the discriminant from the polynomial's coefficients, then interpret the result. For a quadratic, if Δ>0\Delta > 0, there are two distinct real roots. If Δ=0\Delta = 0, there is exactly one repeated real root. If Δ<0\Delta < 0, the two roots are complex conjugates. Higher-degree polynomials have more elaborate discriminant formulas, but the key idea persists: Δ=0\Delta = 0 signals a repeated root.

Worked Example

Problem: Determine the nature of the roots of 2x2+8x+8=02x^2 + 8x + 8 = 0 using the discriminant.
Identify coefficients: Read off aa, bb, and cc from the quadratic.
a=2,b=8,c=8a = 2,\quad b = 8,\quad c = 8
Compute the discriminant: Substitute into the formula.
Δ=82−4(2)(8)=64−64=0\Delta = 8^2 - 4(2)(8) = 64 - 64 = 0
Interpret: Since Δ=0\Delta = 0, the quadratic has exactly one repeated real root.
x=−b2a=−84=−2x = \frac{-b}{2a} = \frac{-8}{4} = -2
Answer: The discriminant is 00, so the equation has one repeated real root at x=−2x = -2.

Why It Matters

The discriminant lets you classify roots without solving the equation, which is especially useful when choosing solution strategies in algebra and precalculus. In physics and engineering, checking Δ\Delta quickly reveals whether a model produces real, physically meaningful solutions.

Common Mistakes

Mistake: Forgetting the −4-4 in b2−4acb^2 - 4ac and computing b2−acb^2 - ac instead.
Correction: The full formula is Δ=b2−4ac\Delta = b^2 - 4ac. The factor of 44 comes from completing the square and cannot be omitted.