mathematics

Rational root theorem

Rational root 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 Rational root theorem rather than just read about it. In short: In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0} with integer coefficients a i ∈ Z {\displaystyle a_{i}\in \mathbb {Z} } and a 0 , a n ≠ 0 {\displaystyle a_{0},a_{n}\neq 0} . Solution…

Key takeaways

Rational root 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 Rational root theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rational root theorem from memory before moving on to harder problems.

Reference excerpt

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation

a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0}

with integer coefficients a i ∈ Z {\displaystyle a_{i}\in \mathbb {Z} } and a 0 , a n ≠ 0 {\displaystyle a_{0},a_{n}\neq 0} . Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution ⁠ x = p q {\displaystyle x={\tfrac {p}{q}}} ⁠ written in lowest terms (that is, p and q are relatively prime), satisfies:

p is an integer factor of the constant term a0, and q is an integer factor of the leading coefficient an. The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is an = 1.

Application The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial (x – r) can be factored out of the polynomial using polynomial long division, resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.

Cubic equation The general cubic equation

a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0}

with integer coefficients has three solutions in the complex plane. If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.

Proofs

Elementary proof Let P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 {\displaystyle P(x)\ =\ a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}} with a 0 , … , a n ∈ Z , a 0 , a n ≠ 0. {\displaystyle a_{0},\ldots ,a_{n}\in \mathbb {Z} ,a_{0},a_{n}\neq 0.} Suppose P(p/q) = 0 for some coprime p, q ∈ ℤ:

P ( p q ) = a n ( p q ) n + a n − 1 ( p q ) n − 1 + ⋯ + a 1 ( p q ) + a 0 = 0. {\displaystyle P\left({\tfrac {p}{q}}\right)=a_{n}\left({\tfrac {p}{q}}\right)^{n}+a_{n-1}\left({\tfrac {p}{q}}\right)^{n-1}+\cdots +a_{1}\left({\tfrac {p}{q}}\right)+a_{0}=0.}

To clear denominators, multiply both sides by qn:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Rational root theorem

Start with the simplest possible case. Write down what Rational root 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 Rational root 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 Rational root 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 Rational root theorem

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

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Rational root theorem in 20 minutes

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

Frequently asked questions

What is Rational root theorem in simple terms?

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0} with integer c…

Why does Rational root 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 Rational root 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 Rational root theorem.