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Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

The theorem that establishes that, using complex numbers, all polynomials can be factored. A generalization of the theorem asserts that any polynomial of degree n has exactly n zeros, counting multiplicity.

 

Fundamental Theorem of Algebra:
A polynomial p(x) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 with degree n at least 1 and with coefficients that may be real or complex must have a factor of the form x – r, where r may be real or complex.

 

 

See also

Factor theorem, polynomial facts

Key Formula

p(x)=an(x−r1)(x−r2)⋯(x−rn)p(x) = a_n(x - r_1)(x - r_2) \cdots (x - r_n)
Where:
  • p(x)p(x) = A polynomial of degree n with real or complex coefficients
  • ana_n = The leading coefficient of the polynomial
  • r1,r2,…,rnr_1, r_2, \ldots, r_n = The n roots (zeros) of the polynomial, which may be real or complex, and some may repeat
  • nn = The degree of the polynomial (must be at least 1)

Worked Example

Problem: Find all roots of the polynomial p(x)=x3−x2+x−1p(x) = x^3 - x^2 + x - 1 and verify that the number of roots matches the degree.
Step 1: The polynomial has degree 3, so the Fundamental Theorem of Algebra tells us there are exactly 3 roots (counting multiplicity) in the complex numbers.
Step 2: Try to find a rational root. Testing x=1x = 1:
p(1)=1−1+1−1=0p(1) = 1 - 1 + 1 - 1 = 0
Step 3: Since x=1x = 1 is a root, (x−1)(x - 1) is a factor. Divide the polynomial by (x−1)(x - 1):
x3−x2+x−1=(x−1)(x2+1)x^3 - x^2 + x - 1 = (x - 1)(x^2 + 1)
Step 4: Now find the roots of x2+1=0x^2 + 1 = 0. Solving gives complex roots:
x2=−1  ⟹  x=i or x=−ix^2 = -1 \implies x = i \text{ or } x = -i
Step 5: Write the full factorization over the complex numbers:
x3−x2+x−1=(x−1)(x−i)(x+i)x^3 - x^2 + x - 1 = (x - 1)(x - i)(x + i)
Answer: The three roots are x=1x = 1, x=ix = i, and x=−ix = -i. This confirms the theorem: a degree-3 polynomial has exactly 3 roots in the complex numbers.

Another Example

Problem: How many roots does p(x)=x4+4p(x) = x^4 + 4 have? Find them.
Step 1: The degree is 4, so the theorem guarantees exactly 4 complex roots (counting multiplicity).
Step 2: Notice this polynomial has no real roots because x4≥0x^4 \geq 0 for all real xx, so x4+4≥4>0x^4 + 4 \geq 4 > 0. All four roots must be non-real complex numbers.
Step 3: Factor using the Sophie Germain identity: x4+4=(x2+2x+2)(x2−2x+2)x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2). Apply the quadratic formula to each factor.
x=−2±4−82=−1±iandx=2±4−82=1±ix = \frac{-2 \pm \sqrt{4-8}}{2} = -1 \pm i \qquad \text{and} \qquad x = \frac{2 \pm \sqrt{4-8}}{2} = 1 \pm i
Answer: The four roots are 1+i1+i, 1−i1-i, −1+i-1+i, and −1−i-1-i. Even though no real roots exist, the theorem ensures all four roots appear in the complex numbers.

Frequently Asked Questions

Does the Fundamental Theorem of Algebra mean every polynomial has a real root?
No. The theorem guarantees roots in the complex numbers, not necessarily real numbers. For example, x2+1=0x^2 + 1 = 0 has no real roots—its roots are ii and −i-i. However, every polynomial with an odd degree and real coefficients will always have at least one real root.
Why does the Fundamental Theorem of Algebra require complex numbers?
If you restrict yourself to real numbers, many polynomials cannot be fully factored into linear factors. The complex numbers form what mathematicians call an algebraically closed field: every non-constant polynomial splits completely into linear factors. Without complex numbers, a polynomial like x2+1x^2 + 1 would have no roots at all.

Fundamental Theorem of Algebra vs. Factor Theorem

The Factor Theorem states that rr is a root of p(x)p(x) if and only if (x−r)(x - r) is a factor—it works one root at a time and does not promise roots exist. The Fundamental Theorem of Algebra goes further: it guarantees that at least one such root rr always exists for any polynomial of degree 1 or higher, ensuring the polynomial can be completely factored into nn linear factors over the complex numbers.

Why It Matters

The Fundamental Theorem of Algebra is the reason you can always solve polynomial equations—you know a solution exists before you start looking for it. It underpins techniques like partial fraction decomposition in calculus and signal processing, where polynomials must be broken into simpler pieces. It also explains why the complex numbers are considered the 'right' number system for polynomial algebra: no further number system extension is needed.

Common Mistakes

Mistake: Thinking a degree-nn polynomial always has nn distinct roots.
Correction: The theorem says nn roots counting multiplicity. For instance, x2−2x+1=(x−1)2x^2 - 2x + 1 = (x-1)^2 has degree 2 but only one distinct root, x=1x = 1, with multiplicity 2.
Mistake: Assuming all roots of a polynomial with real coefficients are real.
Correction: Complex (non-real) roots are perfectly valid. For real-coefficient polynomials, non-real roots come in conjugate pairs (like 3+2i3+2i and 3−2i3-2i), but they are still counted among the nn roots.

Related Terms