Math Lesson 11.3.2 - The Rational Zero (Root) Theorem

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Welcome to our Math lesson on The Rational Zero (Root) Theorem, this is the second lesson of our suite of math lessons covering the topic of Solutions for Polynomial Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Rational Zero (Root) Theorem

The Rational Zero Theorem (otherwise known as the Rational Root Theorem), as its name suggests, is used to find rational solutions when dealing with a polynomial equation (or zeros or roots of a polynomial expression).

A polynomial does not always have zeroes. This means that the corresponding polynomial equations do not always have zeroes. But when they have, these zeroes must be rational numbers. Given that the general form of a polynomial is

or

From the previous tutorials, it is known that all values of the variable x for which P(x) = 0 are the zeroes (roots) of the polynomial. For example, the values -3, -1 and 1 are the zeroes of the polynomial

because

and

The Rational Root Theorem states that all rational roots of a polynomial equation with integer coefficients have the form p/q, where p is a factor of the polynomial constant a₀ and q is a factor of the leading coefficient a_n. The two numbers p and q must be relatively prime.

Proof

First, let's prove that p is a factor of the constant a₀. Indeed, substituting the variable x with the assumed root p/q yields

Multiplying both sides by q_n yields

or

Since p divides each term on the left side, it is a factor of the left side of the above expression. This must be true for the right side as well since both sides are equivalent. This means p is also a factor of the term (-a₀ · q_n). Hence, since p and q are relatively prime and cannot divide each other, the only possibility is that p is a factor of the constant a₀.

Now, let's prove that q is a factor of the leading coefficient a_n. For this, we subtract a_n · p_n from both sides of the polynomial

This yields,

Since q is a factor of all terms on the left side, it is also a factor of the entire expression written on the left. Therefore, it must be a factor of the right side as well, since both sides are equivalent. Given that p and q are relatively prime (they cannot have common factors), then q is a factor of the leading coefficient a_n.

For example, in the polynomial

considered earlier in this tutorial, we have a_n = -1 and a₀ = 3 (n = 3, as it represents the degree of the polynomial). Hence, all roots must be multiples of -1 or 3. Indeed, we proved by substitution that -3, -1 and 1 are roots for the given polynomial.

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1. Rational Zero Root Theorem Feedback. Helps other - Leave a rating for this rational zero root theorem (see below)
2. Polynomials Math tutorial: Solutions for Polynomial Equations. Read the Solutions for Polynomial Equations math tutorial and build your math knowledge of Polynomials
3. Polynomials Video tutorial: Solutions for Polynomial Equations. Watch or listen to the Solutions for Polynomial Equations video tutorial, a useful way to help you revise when travelling to and from school/college
4. Polynomials Revision Notes: Solutions for Polynomial Equations. Print the notes so you can revise the key points covered in the math tutorial for Solutions for Polynomial Equations
5. Polynomials Practice Questions: Solutions for Polynomial Equations. Test and improve your knowledge of Solutions for Polynomial Equations with example questins and answers
6. Check your calculations for Polynomials questions with our excellent Polynomials calculators which contain full equations and calculations clearly displayed line by line. See the Polynomials Calculators by iCalculator™ below.
7. Continuing learning polynomials - read our next math tutorial: Rational Expressions

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