Complex Zeros of a Polynomial: Conjugate Pairs and Linear Factorization

The zeros lesson finds where a polynomial crosses the -axis. But never crosses it, and crosses only once despite being a cubic. The zeros that the graph cannot show are complex zeros, and three theorems describe them completely: every polynomial has as many zeros as its degree once complex numbers are allowed; the complex ones come in conjugate pairs; and the polynomial factors all the way down into linear pieces. Together they turn “how many zeros does this have?” from a guess into a count.

What Is a Complex Zero?

A zero of a polynomial is any input that makes the output zero. If you only allow real inputs, some polynomials have fewer zeros than their degree, or none at all. Allowing inputs from the complex numbers — numbers of the form with — fills the gaps.

For :

So and are zeros of . They are called complex zeros or, when the real part is zero as here, imaginary zeros. The graph of sits entirely above the axis: complex zeros are invisible on a graph, because a graph only plots real .

A real zero is a complex zero whose imaginary part happens to be . So “complex zeros” includes the real ones — when a question says find all complex zeros, it means find every zero, real or not.

The Fundamental Theorem of Algebra

Every polynomial of degree has at least one complex zero. Applying that once, dividing out the corresponding factor, and repeating gives the useful form:

A polynomial of degree has exactly complex zeros, counting each zero as many times as its multiplicity.

That is the complete answer to how many solutions does the equation have over the complex number system: as many as the degree. Some may be real, some may repeat, but the count always comes out to .

The fourth row is the cubic from the introduction: one crossing on the graph, two zeros hiding off it.

The Conjugate Pairs Theorem

If a polynomial has real coefficients and is a zero, then its conjugate is also a zero.

The reason is short. Take the conjugate of both sides of . Conjugation respects addition and multiplication, and it leaves real numbers alone — so every real coefficient stays put, every becomes , and the equation becomes .

Two consequences that get used constantly:

• Non-real zeros always come in pairs. A real-coefficient polynomial cannot have an odd number of non-real zeros. So a cubic has either three real zeros or one real zero plus a complex pair — never two real and one complex.
• Every odd-degree polynomial has at least one real zero. With odd and the non-real zeros paired off, at least one zero is left over unpaired, and it must be real. (The graph says the same thing: odd degree means the tails go opposite ways, so the curve has to cross.)

The theorem needs real coefficients. has the single zero and no partner, because its constant term is not real.

The Linear Factorization Theorem

Putting the two theorems together: a degree- polynomial with leading coefficient and zeros (listed with multiplicity) factors completely as

Over the complex numbers, every polynomial splits into linear factors — there is no such thing as an irreducible quadratic once is allowed. Over the reals, each conjugate pair multiplies back into a real quadratic:

which has a negative discriminant and cannot be factored further with real numbers. So the real factorization of any real-coefficient polynomial is a product of linear factors (one per real zero) and irreducible quadratics (one per conjugate pair):

How to Find the Complex Zeros of a Polynomial

The strategy is to reduce the polynomial to a quadratic, because the quadratic formula handles complex zeros automatically — a negative discriminant simply produces a pair with an in it.

1. Find the real zeros first. Use the rational root theorem to list candidates and synthetic division to test them, dividing each one out as you find it.
2. Stop when the quotient is quadratic. If its discriminant is negative, the last two zeros are a conjugate pair from the quadratic formula.
3. If you are given a complex zero, write down its conjugate too, multiply the pair into the real quadratic , and divide the polynomial by that quadratic. The quotient holds the remaining zeros.
4. Count. The zeros you have found, with multiplicity, should number exactly the degree.

Worked Example: Reducing to a Quadratic

Find all complex zeros of .

Step 1 — rational candidates. Constant , leading coefficient : candidates .

Step 2 — test by synthetic division:

The bottom row is with remainder , so is a zero and the quotient is .

Step 3 — solve the quadratic. gives , so

Step 4 — count. Three zeros for a cubic: , , . ✓

The graph confirms it — one crossing at and no others.

Worked Example: Given One Complex Zero

is a zero of . Find the remaining zeros.

Step 1 — the conjugate is also a zero, because the coefficients are real.

Step 2 — multiply the pair into a real quadratic, using with , :

Step 3 — divide by with long division:

(Check the leading terms: . Check the constants: . Remainder .)

Step 4 — factor the quotient:

Step 5 — count. Zeros: , , , — four for a quartic. ✓

Worked Example: A Quadratic in Disguise

Find all complex zeros of .

Step 1 — recognise the pattern. Only even powers appear, so substitute :

Step 2 — return to :

Step 3 — each factor gives a conjugate pair:

Step 4 — count. Four zeros, all imaginary, for a quartic. ✓ The graph of this polynomial never touches the -axis at all — consistent with a quartic (both tails up) whose minimum value is .

Common Mistakes to Avoid

• Reporting only the real zeros. “Find all complex zeros of a cubic” expects three answers. If you have one, the other two are a conjugate pair waiting in the quadratic quotient.
• Forgetting the conjugate. Given that is a zero, comes free. Dividing by alone means complex arithmetic in the division for no reason.
• Applying the conjugate pairs theorem to complex coefficients. It only holds when every coefficient is real. has the double zero and no .
• Expanding a conjugate pair term by term. is a difference of squares in disguise: . Four-term expansion invites a sign error on .
• Saying a polynomial “has no zeros”. Over the reals, has no zeros; over the complex numbers it has two. Be clear which you mean — an exam question about complex zeros always means the second.
• Losing count of multiplicity. has four zeros — — not three. The repeated counts twice.

Practice Problems

Work each one before opening the answer.

Problem 1. How many complex zeros does have, counting multiplicity?

Problem 2. Find all complex zeros of .

Problem 3. A polynomial with real coefficients has zeros and . What is the smallest possible degree, and what are all its zeros?

Problem 4. Find all complex zeros of .

Problem 5. Find all complex zeros of .

Problem 6. is a zero of . Find all four zeros.

Complex zeros complete the story that real zeros start: exactly of them, paired off when the coefficients are real, and one linear factor each. Running the construction backwards — building a polynomial from a list of zeros that includes a complex pair — is the writing a polynomial from its zeros lesson, and the arithmetic of itself is covered under complex numbers.