Rational Root Theorem

What You’ll Learn

In this lesson you’ll learn the Rational Root Theorem and how to use it systematically to find possible rational roots of polynomial equations.

The Concept

The Rational Root Theorem provides a list of all possible rational roots of a polynomial equation with integer coefficients.

If a polynomial with integer coefficients has a rational root p/q (in lowest terms), then:

• p is a factor of the constant term (the last number)
• q is a factor of the leading coefficient (the first number)

This theorem doesn’t tell you which roots are actual solutions. It only gives you a finite list of candidates to test. You still need to plug them in or use synthetic division to check.

Use the Rational Root Theorem on 2x³ − 5x² − 4x + 3 = 0:

How the Rational Root Theorem generates candidates

The diagram shows the process visually for the polynomial 2x³ − 5x² − 4x + 3. The blue 2 (leading coefficient) gives the q factors: ±1, ±2. The green 3 (constant term) gives the p factors: ±1, ±3. Dividing every p by every q produces the gold list of candidates: ±1, ±3, ±1/2, ±3/2. These are the only rational numbers that could possibly be roots.

Steps to use it:

1. List all positive and negative factors of the constant term (these are the possible p values).
2. List all positive and negative factors of the leading coefficient (these are the possible q values).
3. Form all possible fractions p/q and reduce them.
4. Test each candidate by plugging it into the polynomial (or using synthetic division). If the result is 0, it’s a root.

Worked Example

Use the Rational Root Theorem to factor 2x³ − 5x² − 4x + 3 = 0.

1. List the possible rational roots

Constant term = 3, so factors of 3 are: ±1, ±3.

Leading coefficient = 2, so factors of 2 are: ±1, ±2.

All possible rational roots (p/q): ±1, ±3, ±1/2, ±3/2.

2. Test candidates

Test x = 1:

$$2(1)^3 - 5(1)^2 - 4(1) + 3 = 2 - 5 - 4 + 3 = -4 \neq 0$$

Not a root. Try x = 3:

$$2(3)^3 - 5(3)^2 - 4(3) + 3 = 54 - 45 - 12 + 3 = 0 \;\checkmark$$

x = 3 is a root, so (x − 3) is a factor.

3. Use synthetic division to divide 2x³ − 5x² − 4x + 3 by (x − 3)

3	2	−5	−4	3
		6	3	−3
	2	1	−1	0

The quotient is 2x² + x − 1.

4. Factor the remaining quadratic

$$2x^2 + x - 1 = (2x - 1)(x + 1)$$

Final factored form:

$$2x^3 - 5x^2 - 4x + 3 = (x - 3)(2x - 1)(x + 1)$$

The roots are x = 3, x = 1/2, and x = −1.

y = 2x³ − 5x² − 4x + 3 — the roots match our Rational Root Theorem results

The graph confirms our algebra. The blue cubic crosses the x-axis at exactly the three roots we found: x = −1, x = 1/2, and x = 3.

Real-World Application

The Rational Root Theorem is useful whenever you need to solve a polynomial equation and don’t have a graph handy:

• Finding break-even points in business profit models
• Solving engineering equations for dimensions or forces
• Analyzing motion equations in physics
• Determining equilibrium points in economics

Without this theorem, you’d have to guess randomly. The Rational Root Theorem narrows the search to a manageable list.

You’ve Got This

The Rational Root Theorem is a systematic way to start. It gives you a checklist of candidates instead of random guessing. Not every candidate will work, but the list is always finite. Test them one at a time (start with the easy ones like ±1, ±2), and once you find one root, use synthetic division to reduce the degree. Then factor what’s left.

Quiz

For $3x^3 + 2x^2 - x - 6 = 0$, which are possible rational roots?

• A.$\pm 1, \pm 2, \pm 3$
• B.$\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}$
• C.$\pm 1, \pm 6$
• D.$\pm 3, \pm 6$

In the Rational Root Theorem, p is a factor of the:

• A.Middle term
• B.Leading coefficient
• C.Degree
• D.Constant term

If you test a candidate root and get 0, that means:

• A.It is a root of the polynomial
• B.It is not a root
• C.The polynomial has no roots
• D.You need to test more values

After finding one rational root, the best next step is:

• A.Use the quadratic formula on the original
• B.Test all remaining candidates
• C.Synthetic division to reduce the degree
• D.Graph the polynomial

List all possible rational roots of $2x^3 - x^2 + 4x - 2 = 0$.

• A.$\pm 1, \pm 2$
• B.$\pm 1, \pm 2, \pm \frac{1}{2}$
• C.$\pm 1, \pm 2, \pm 4$
• D.$\pm \frac{1}{2}, \pm \frac{1}{4}$