Roots & Zeros
A polynomial's zeros are the inputs that make it zero — its roots. The Fundamental Theorem of Algebra guarantees a degree- polynomial has exactly roots, counting multiplicity and complex ones.
When coefficients are real, complex roots travel in conjugate pairs: if is a root, so is . That pairing is the Conjugate Root Theorem.
Counting all the roots
Every degree- polynomial has exactly roots when you count repeats and complex values. Some may be real, some complex.
So a cubic always has three roots total — perhaps one real and two complex, or three real.
The Conjugate Root Theorem
For a polynomial with real coefficients, complex roots occur in conjugate pairs. A root of forces to be a root as well.
The same holds for irrational roots involving square roots: pairs with . This lets you name a partner root for free.
Worked examples
Example 1: the conjugate partner
A polynomial with real coefficients has as a zero. Which value must also be a zero?
Answer:
Example 2: total root count
How many roots does a degree-5 polynomial have, counting multiplicity?
Answer:
Example 3: an irrational conjugate partner
A polynomial with rational coefficients has as a zero. Which value must also be a zero?
Answer:
Try one yourself
Common questions
How many roots does a polynomial have?
Exactly its degree, when you count multiplicity and include complex roots. A degree-4 polynomial has four roots total.
Why do complex roots come in pairs?
For real coefficients, the Conjugate Root Theorem forces it: if is a root, so is .
Does the pairing apply to irrational roots?
Yes, for roots with square roots: pairs with when the coefficients are rational.
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