mathematics

Fundamental theorem of algebra

Fundamental theorem of algebra is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Fundamental theorem of algebra rather than just read about it. In short: The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Key takeaways

Fundamental theorem of algebra belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Fundamental theorem of algebra to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Fundamental theorem of algebra from memory before moving on to harder problems.

Reference excerpt

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, it is not fundamental for modern algebra; it was named when algebra was synonymous with the theory of equations. Additionally, the theorem cannot be proven in Algebra: transcendental methods such as, for example, topology and the theory of holomorphic functions of one complex variable are required to prove the theorem.

History Peter Roth, in his book Arithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds "unless the equation is incomplete", where "incomplete" means that at least one coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation x 4 = 4 x − 3 , {\displaystyle x^{4}=4x-3,} although incomplete, has four solutions (counting multiplicities): 1 (twice), − 1 + i 2 , {\displaystyle -1+i{\sqrt {2}},} and − 1 − i 2 . {\displaystyle -1-i{\sqrt {2}}.}

As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degrees are either 1 or 2. However, in 1702 Leibniz erroneously said that no polynomial of the type x4 + a4 (with a real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x4 − 4x3 + 2x2 + 4x + 4, but he got a letter from Euler in 1742 in which it was shown that this polynomial is equal to

( x 2 − ( 2 + α ) x + 1 + 7 + α ) ( x 2 − ( 2 − α ) x + 1 + 7 − α ) , {\displaystyle \left(x^{2}-(2+\alpha )x+1+{\sqrt {7}}+\alpha \right)\left(x^{2}-(2-\alpha )x+1+{\sqrt {7}}-\alpha \right),}

with α = 4 + 2 7 . {\displaystyle \alpha ={\sqrt {4+2{\sqrt {7}}}}.}

Euler also pointed out that

x 4 + a 4 = ( x 2 + a 2 ⋅ x + a 2 ) ( x 2 − a 2 ⋅ x + a 2 ) . {\displaystyle x^{4}+a^{4}=\left(x^{2}+a{\sqrt {2}}\cdot x+a^{2}\right)\left(x^{2}-a{\sqrt {2}}\cdot x+a^{2}\right).}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Fundamental theorem of algebra

Start with the simplest possible case. Write down what Fundamental theorem of algebra claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Fundamental theorem of algebra before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Fundamental theorem of algebra ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Fundamental theorem of algebra

In research: Fundamental theorem of algebra appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Fundamental theorem of algebra in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Fundamental theorem of algebra is common in secondary-school and first-year university syllabi. It links to neighbouring topics Field theory, Theorems about polynomials, Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Fundamental theorem of algebra outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Fundamental theorem of algebra in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Fundamental theorem of algebra means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Fundamental theorem of algebra out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Fundamental theorem of algebra in simple terms?

Why does Fundamental theorem of algebra matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Fundamental theorem of algebra?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Fundamental theorem of algebra.