mathematics

Polynomial remainder theorem

Polynomial remainder theorem 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 Polynomial remainder theorem rather than just read about it. In short: In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that, for every number r {\displaystyle r} , any polynomial f ( x ) {\displaystyle f(x)} is the sum of f ( r ) {\displaystyle f(r)} and the product of x − r {\displaystyle x-r} and a polynomial in x {\displaystyle x} of a degree one less than the degre…

Key takeaways

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

Reference excerpt

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that, for every number r {\displaystyle r} , any polynomial f ( x ) {\displaystyle f(x)} is the sum of f ( r ) {\displaystyle f(r)} and the product of x − r {\displaystyle x-r} and a polynomial in x {\displaystyle x} of a degree one less than the degree of f {\displaystyle f} . In particular, f ( r ) {\displaystyle f(r)} is the remainder of the Euclidean division of f ( x ) {\displaystyle f(x)} by x − r {\displaystyle x-r} , and x − r {\displaystyle x-r} is a divisor of f ( x ) {\displaystyle f(x)} if and only if f ( r ) = 0 {\displaystyle f(r)=0} , a property known as the factor theorem.

Examples

Example 1 Let f ( x ) = x 3 − 12 x 2 − 42 {\displaystyle f(x)=x^{3}-12x^{2}-42} . Polynomial division of f ( x ) {\displaystyle f(x)} by ( x − 3 ) {\displaystyle (x-3)} gives the quotient x 2 − 9 x − 27 {\displaystyle x^{2}-9x-27} and the remainder − 123 {\displaystyle -123} . By the polynomial remainder theorem, f ( 3 ) = − 123 {\displaystyle f(3)=-123} .

Example 2 Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial f ( x ) = a x 2 + b x + c {\displaystyle f(x)=ax^{2}+bx+c} by using algebraic manipulation:

f ( x ) − f ( r ) = a x 2 + b x + c − ( a r 2 + b r + c ) = a ( x 2 − r 2 ) + b ( x − r ) = a ( x − r ) ( x + r ) + b ( x − r ) = ( x − r ) ( a x + a r + b ) {\displaystyle {\begin{aligned}f(x)-f(r)&=ax^{2}+bx+c-(ar^{2}+br+c)\\&=a(x^{2}-r^{2})+b(x-r)\\&=a(x-r)(x+r)+b(x-r)\\&=(x-r)(ax+ar+b)\end{aligned}}}

So,

f ( x ) = ( x − r ) ( a x + a r + b ) + f ( r ) , {\displaystyle f(x)=(x-r)(ax+ar+b)+f(r),}

which is exactly the formula of Euclidean division. The generalization of this proof to any degree is given below in § Direct proof.

Proofs

Using Euclidean division The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Polynomial remainder theorem

Start with the simplest possible case. Write down what Polynomial remainder theorem 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 Polynomial remainder theorem 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 Polynomial remainder theorem 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 Polynomial remainder theorem

In research: Polynomial remainder theorem 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 Polynomial remainder theorem 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: Polynomial remainder theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems about polynomials, so understanding it makes those chapters shorter.
In everyday life: Look for Polynomial remainder theorem 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 Polynomial remainder theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Polynomial remainder theorem 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 Polynomial remainder theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Polynomial remainder theorem in simple terms?

Why does Polynomial remainder theorem 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 Polynomial remainder theorem?

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 Polynomial remainder theorem.