← College Algebra

College Algebra

Complex Zeros and the Fundamental Theorem

Lesson

The Fundamental Theorem of Algebra is the guarantee that polynomial equations always have solutions — if you allow complex numbers.

The theorem

A polynomial of degree nn has exactly nn complex zeros (counting multiplicity).

Some of those zeros may be real, some may be complex. Their total is always nn.

Complex conjugate pairs

If a polynomial has real coefficients and a+bia + bi is a zero, then so is a−bia - bi. Complex zeros always come in conjugate pairs for real polynomials.

Consequence: a polynomial of degree nn has either 0, 2, 4, ... complex (non-real) zeros — always an even number.

Worked example 1

x2+9=0x^2 + 9 = 0
x2=−9  ⟹  x=±3ix^2 = -9 \implies x = \pm 3i

Two complex zeros: 3i3i and −3i-3i (a conjugate pair).

Worked example 2 — quadratic formula on a negative discriminant

x2−2x+5=0x^2 - 2x + 5 = 0
x=2±4−202=2±4i2=1±2ix = \frac{2 \pm \sqrt{4 - 20}}{2} = \frac{2 \pm 4i}{2} = 1 \pm 2i

Conjugate pair: 1+2i1 + 2i and 1−2i1 - 2i. Real part is 11, imaginary part is ±2\pm 2.

Counting non-real zeros

For a polynomial of degree nn with rr distinct real zeros, the number of non-real (complex) zeros must be an even number that gets the total up to nn.

Example: degree 6 with 4 real zeros must have 2 non-real zeros.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

How many complex zeros does x2−4 have?\text{How many complex zeros does } x^2 - 4 \text{ have?}

Problem 2

How many complex zeros does x3+1 have?\text{How many complex zeros does } x^3 + 1 \text{ have?}

Problem 3

How many complex zeros does x5−1 have?\text{How many complex zeros does } x^5 - 1 \text{ have?}

Problem 4

Imaginary part of the conjugate of 3+2i\text{Imaginary part of the conjugate of } 3 + 2i

Practice

Standard problems matching the lesson.

Problem 5

Number of complex zeros of a degree-7 polynomial\text{Number of complex zeros of a degree-7 polynomial}

Problem 6

Real part of the conjugate of 5+i\text{Real part of the conjugate of } 5 + i

Problem 7

Imaginary part of the conjugate of 5+i\text{Imaginary part of the conjugate of } 5 + i

Problem 8

Imaginary part of the positive-imaginary root of x2+25=0\text{Imaginary part of the positive-imaginary root of } x^2 + 25 = 0

Problem 9

Imaginary part of the positive root of x2+100=0\text{Imaginary part of the positive root of } x^2 + 100 = 0

Problem 10

Imaginary part of the positive root of x2+1=0\text{Imaginary part of the positive root of } x^2 + 1 = 0

Problem 11

Imaginary part of the positive root of x2+16=0\text{Imaginary part of the positive root of } x^2 + 16 = 0

Problem 12

Real part of the conjugate of 2−3i\text{Real part of the conjugate of } 2 - 3i

Problem 13

Imaginary part of the conjugate of −1+4i\text{Imaginary part of the conjugate of } -1 + 4i

Problem 14

x^2 + 8x + 25 = 0, imag part of positive root?

Problem 15

x^2 - 2x + 5 = 0, real part?

Problem 16

x^2 - 2x + 5 = 0, imag part?

Problem 17

Degree-4 with zeros 2, -1, 3+i. Imag part of 4th?

Problem 18

Number of complex zeros of a degree-10 polynomial\text{Number of complex zeros of a degree-10 polynomial}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

x^2 + 6x + 13 = 0, real part?

Problem 20

x^2 + 6x + 13 = 0, imag part?

Problem 21

x^2 + 2x + 10 = 0, imag part?

Problem 22

x^2 - 4x + 13 = 0, real part?

Problem 23

x^2 - 4x + 13 = 0, imag part?

Problem 24

Degree-5 with zeros 1, 2, 3, 4+i. Non-real zeros total?

Problem 25

Degree-6 with 4 real zeros. Non-real zeros?

Ask the tutor

Stuck on a concept? Want another example? Ask anything about this topic.

Type your own question below, or tap one of the suggestions. The tutor can re-explain the lesson, work through a specific problem with you, generate fresh practice tuned to where you are, or check your reasoning.

Quiz

Test yourself on this topic →

10 questions, no hints. About 5 minutes.