Finding Zeros: Rational Root Theorem

By the end of this lesson, you will be able to use the Rational Root Theorem to list all possible rational zeros, test candidates with the Factor Theorem, divide to reduce degree, and find all zeros of higher-degree polynomials.

The Big Idea

Finding zeros of degree 3+ polynomials requires a systematic approach. The Rational Root Theorem provides a finite list of rational candidates — you don't have to guess randomly. After testing, each confirmed zero lets you synthetic divide to reduce degree, eventually reaching a quadratic you can solve exactly.

The Rational Root Theorem

If $p(x) = a_n x^n + \cdots + a_0$ has integer coefficients, then every rational zero has the form:

$$x = \pm\frac{p}{q}$$

where:

• $p$ divides the constant term $a_0$
• $q$ divides the leading coefficient $a_n$

This gives a finite list of candidates. None are guaranteed to be actual zeros — you must test each.

Descartes' Rule of Signs (Complement)

Positive zeros: count sign changes in $p(x)$; the number of positive real zeros equals this count or is less by an even number.

Negative zeros: count sign changes in $p(-x)$; same rule.

This narrows which candidates to test first.

Example 1 — Full Rational Root Procedure

Find all zeros of $p(x) = 2x^3 - x^2 - 7x + 6$.

Step 1: Constant term = 6, divisors: $\pm1, \pm2, \pm3, \pm6$.

Leading coeff. = 2, divisors: $\pm1, \pm2$.

Candidates: $\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}$

Step 2: Test via $p(r) = 0$.

$p(1) = 2 - 1 - 7 + 6 = 0$ ✓ → $x = 1$ is a zero.

Step 3: Synthetic divide by $r = 1$:

1 | 2   -1   -7    6
  |      2    1   -6
  |------------------
    2    1   -6    0

Quotient: $2x^2 + x - 6$

Step 4: Factor the quotient:

$2x^2 + x - 6 = (2x - 3)(x + 2)$

Zeros: $x = \frac{3}{2}$ and $x = -2$

All zeros: $x = 1, \frac{3}{2}, -2$

Example 2 — Repeated Zero

$p(x) = x^4 - 5x^3 + 9x^2 - 7x + 2$.

Test $x = 1$: $1 - 5 + 9 - 7 + 2 = 0$ ✓

After dividing: quotient is $x^3 - 4x^2 + 5x - 2$.

Test $x = 1$ again on quotient: $1 - 4 + 5 - 2 = 0$ ✓ (double zero!)

After dividing: $x^2 - 3x + 2 = (x-1)(x-2)$