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Conjugate Pair Theorem — Definition, Proof & Examples

Conjugate Pair Theorem

An assertion about the complex zeros of any polynomial which has real numbers as coefficients.

Theorem:

If a polynomial

p(x) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0

has real coefficients, then any complex zeros occur in conjugate pairs. That is, if a + bi is a zero then so is a – bi and vice-versa.
Example:

2 – 3i is a zero of p(x) = x3 – 3x2 + 9x + 13 as shown here:

p(2 – 3i) = (2 – 3i)3 – 3(2 – 3i)2 + 9(2 – 3i) + 13
              = (–46 – 9i) – 3(–5 – 12i) + (18 – 27i) + 13
              = –46 – 9i + 15 + 36i + 18 – 27i + 13
              = 0.

By the conjugate pair theorem, 2 + 3i is also a zero of p(x).

p(2 + 3i) = (2 + 3i)3 – 3(2 + 3i)2 + 9(2 + 3i) + 13
              = (–46 + 9i) – 3(–5 + 12i) + (18 + 27i) + 13
              = –46 + 9i + 15 – 36i + 18 + 27i + 13
              = 0.

See also

Complex conjugate, polynomial facts

Worked Example

Problem: You know that 1+2i1 + 2i is a zero of the polynomial p(x)=x3−5x2+11x−15p(x) = x^3 - 5x^2 + 11x - 15. Find all the zeros of p(x)p(x).
Step 1: Apply the Conjugate Pair Theorem. Since all coefficients of p(x)p(x) are real and 1+2i1 + 2i is a zero, the conjugate 1−2i1 - 2i must also be a zero.
Zeros so far: 1+2i and 1−2i\text{Zeros so far: } 1+2i \text{ and } 1-2i
Step 2: Form the quadratic factor from these two conjugate zeros. Multiply the corresponding linear factors together.
(x−(1+2i))(x−(1−2i))=(x−1)2−(2i)2=x2−2x+1+4=x2−2x+5\bigl(x-(1+2i)\bigr)\bigl(x-(1-2i)\bigr) = (x-1)^2 - (2i)^2 = x^2 - 2x + 1 + 4 = x^2 - 2x + 5
Step 3: Divide p(x)p(x) by this quadratic factor to find the remaining factor.
x3−5x2+11x−15x2−2x+5=x−3\frac{x^3 - 5x^2 + 11x - 15}{x^2 - 2x + 5} = x - 3
Step 4: Set the remaining factor equal to zero to find the third zero.
x−3=0  ⟹  x=3x - 3 = 0 \implies x = 3
Answer: The three zeros of p(x)p(x) are 1+2i1 + 2i, 1−2i1 - 2i, and 33.

Another Example

Problem: A polynomial with real coefficients has degree 4 and its known zeros are 33, −1-1, and 4+i4 + i. Find the fourth zero.
Step 1: Identify which zeros are complex. The zeros 33 and −1-1 are real, so they don't require conjugate partners. The zero 4+i4 + i is complex (it has a nonzero imaginary part).
Step 2: Apply the Conjugate Pair Theorem. Since the polynomial has real coefficients and 4+i4 + i is a zero, its conjugate must also be a zero.
Fourth zero=4−i\text{Fourth zero} = 4 - i
Step 3: Verify the count. A degree-4 polynomial has exactly 4 zeros (counting multiplicity): 33, −1-1, 4+i4+i, and 4−i4-i. This matches.
Answer: The fourth zero is 4−i4 - i.

Frequently Asked Questions

Does the Conjugate Pair Theorem apply to polynomials with complex (non-real) coefficients?
No. The theorem requires all coefficients to be real numbers. If a polynomial has non-real coefficients, such as p(x)=x−(2+3i)p(x) = x - (2 + 3i), a complex zero can appear without its conjugate. In that example, 2+3i2 + 3i is a zero but 2−3i2 - 3i is not.
Can a polynomial with real coefficients and odd degree have all complex (non-real) zeros?
No. By the Conjugate Pair Theorem, complex zeros come in pairs, so the number of non-real zeros is always even. An odd-degree polynomial has an odd total number of zeros, which means at least one zero must be real.

Conjugate Pair Theorem vs. Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra says every polynomial of degree n≥1n \geq 1 has exactly nn zeros in the complex numbers (counting multiplicity). The Conjugate Pair Theorem adds a structural constraint: when the coefficients are all real, those complex zeros are symmetric — they appear in conjugate pairs. The Fundamental Theorem tells you how many zeros exist; the Conjugate Pair Theorem tells you how they are related to each other.

Why It Matters

The Conjugate Pair Theorem lets you find missing zeros efficiently. If you discover one complex zero of a real-coefficient polynomial, you immediately know another zero for free. This also explains why quadratic factors with no real roots always have the form x2+bx+cx^2 + bx + c with a negative discriminant — the two complex roots are always conjugates of each other.

Common Mistakes

Mistake: Applying the theorem to polynomials with non-real coefficients.
Correction: The theorem only applies when every coefficient of the polynomial is a real number. Always check this condition before concluding that a conjugate is also a zero.
Mistake: Thinking that real zeros also need conjugate pairs.
Correction: A real number like 55 is its own conjugate (5+0i5 + 0i and 5−0i5 - 0i are the same number). The theorem is only informative for zeros with a nonzero imaginary part.

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