Complex Zeros and Conjugate Pairs

A parabola that never crosses the x-axis still has two zeros, they just aren't real numbers. The Fundamental Theorem of Algebra guarantees a degree-n polynomial has exactly n zeros once you count multiplicity, and it does not require those zeros to be visible on a real graph. When a quadratic's discriminant is negative, the quadratic formula still spits out two answers, x = a + bi and x = a - bi, using sqrt of a negative number. Those are the two zeros. The graph missing the x-axis just means both zeros left the real number line.

Non-real zeros arrive in pairs because the polynomial's coefficients are real. If you plug a + bi into the polynomial and it equals zero, swapping every i for -i throughout the whole computation also gives zero, because every coefficient multiplying those terms is real and unaffected by the swap. So a - bi must be a zero too. This is the conjugate pairs rule: non-real zeros of a real-coefficient polynomial always travel together, never alone.

This connects directly to counting zeros by degree. A quartic has 4 zeros total. If 2 are real, the remaining 2 must be a non-real conjugate pair, not one lone non-real zero, since that would break the pairing rule. Multiplicity matters too: a repeated real zero counts twice toward the total. Once you add up real zeros with their multiplicities and non-real zeros in their conjugate pairs, the count always lands exactly on the degree.