Math Practice
Over the complex numbers every degree-n polynomial has exactly n zeros, and the non-real ones arrive in conjugate pairs. These twelve problems drill the counting, the pairing, and the routine of reducing a polynomial to a quadratic whose discriminant is negative.
How many complex zeros does
The Fundamental Theorem of Algebra: a degree-6 polynomial has exactly 6 complex zeros counting multiplicity, regardless of how many are real.
Find the zeros of
A polynomial with real coefficients has
The conjugate pairs theorem: with real coefficients, non-real zeros come in conjugate pairs, so
Can a cubic polynomial with real coefficients have exactly two non-real zeros and one real zero? Answer "yes" or "no".
Yes. The two non-real zeros form a conjugate pair, and the third zero is real — for example
Find the zeros of
Discriminant
What real quadratic (leading coefficient
Use
How many real zeros does
Find all zeros of
Group:
Find all zeros of
Quadratic in
The conjugate
A degree-
Each given non-real zero brings its conjugate:
Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work
The complex zeros lesson rests on three facts: a degree-
Real zeros plus non-real zeros, each counted as many times as it repeats, always add to the degree.
The pair multiplies back into the real quadratic
Once a polynomial is reduced to a quadratic with a negative discriminant, the quadratic formula produces both complex zeros at once.
Exactly as many as its degree, counting multiplicity. A degree-4 polynomial has four complex zeros, some of which may be real and some of which may repeat.
A real zero is where the graph crosses or touches the x-axis. A complex zero a + bi with b ≠ 0 does not appear on the graph at all. Technically real numbers are complex numbers too, so 'all complex zeros' means every zero, real or not.
Because the polynomial's coefficients are real. Conjugating both sides of P(z) = 0 leaves real coefficients unchanged and turns z into its conjugate, so the conjugate is also a zero.
Zeros p ± qi give the real quadratic (x − p)² + q² = x² − 2px + (p² + q²). For 3 ± 2i that is x² − 6x + 13.
Write down its conjugate, multiply the pair into a real quadratic, divide the polynomial by that quadratic, and solve the quotient.
No. The non-real zeros pair off, which uses up an even number, so an odd degree always leaves at least one zero that must be real.