Descartes' Rule of Signs

The rational root theorem produces a list of candidate zeros, and for a polynomial like that list has sixteen entries. Descartes’ rule of signs trims it before you test anything. By counting the sign changes in the coefficients of , and then of , it tells you the possible numbers of positive and negative real zeros — and if it says there are no negative zeros, half the candidates go straight in the bin. It never finds a zero itself, but it tells you how many of each kind to look for and when to stop.

What a Sign Change Is

Write the polynomial in standard form, then read the signs of the coefficients left to right, skipping any missing powers. A sign change (or variation in sign) is any place where a is followed by a or a by a .

For , the signs are

and there are three changes: , , . The missing term is simply ignored; zero coefficients never count as a change and never block one.

Descartes’ Rule of Signs

Let be a polynomial with real coefficients and a non-zero constant term.

Positive real zeros. The number of positive real zeros of is equal to the number of sign changes in , or less than that by an even number.

Negative real zeros. The number of negative real zeros of is equal to the number of sign changes in , or less than that by an even number.

Each zero is counted with its multiplicity. So three sign changes in means or positive zeros; two sign changes means or ; one sign change means exactly ; and no sign changes means none.

The “less by an even number” clause is the conjugate pairs theorem showing through: the zeros that fail to be real go missing two at a time.

How to Use It: The Three Steps

Step 1 — count the sign changes in . That gives the possibilities for positive zeros.

Step 2 — form and count its sign changes. Replace every with . Odd powers flip sign; even powers and the constant do not. Then count. That gives the possibilities for negative zeros.

Step 3 — build the table. List every combination of (positive, negative) allowed by Steps 1 and 2. Each row’s non-real count is what is left over: degree minus positive minus negative. Since non-real zeros come in pairs, that leftover must be even — and it always will be, which is a useful check on your counting.

For :

Signs : one change. So exactly one negative zero. With three changes in :

Two possible profiles for the four zeros, and either way exactly one of the eight negative rational-root candidates can be a zero.

Two Things the Rule Needs

A non-zero constant term. If has no constant term, then is a zero — neither positive nor negative. Factor out the highest power of first and apply the rule to what remains. For , factor to , then apply the rule to : one sign change, so one positive zero; again, so one negative zero. Together with , that accounts for all three.

Real coefficients. The rule relies on non-real zeros pairing off, which only happens when the coefficients are real.

Worked Example: Descartes’ Rule of Signs Step by Step

Use Descartes’ rule of signs on , then say how many zeros must be non-real.

Step 1 — sign changes in . Coefficients (the term is missing):

Four changes. Positive zeros: , , or .

Step 2 — sign changes in . Odd powers flip:

Signs : one change. Negative zeros: exactly .

Step 3 — the table. Degree :

Every row has an even number of non-real zeros. ✓

Answer. has exactly one negative real zero, an even number of positive real zeros, and therefore , or non-real zeros. Without more work that is as far as the rule goes — but it already guarantees at least one real zero, which for an odd degree the graph confirms.

Worked Example: No Sign Changes

Show that has no real zeros.

Step 1 — . Signs : no changes. No positive zeros.

Step 2 — . Every power is even, so , the same polynomial: no changes. No negative zeros.

Step 3 — zero is not a zero, since the constant term is .

Answer. No real zeros at all; all four zeros are non-real. (In fact , with zeros and .) Descartes’ rule settled the question without solving anything.

Worked Example: Trimming the Candidate List

Find all real zeros of .

Step 1 — rational candidates. Constant , leading coefficient : . Eight candidates.

Step 2 — Descartes on . Signs : three changes. Positive zeros: or .

Step 3 — Descartes on .

Signs : no changes. No negative zeros. The four negative candidates are eliminated without testing.

Step 4 — test the positive candidates with synthetic division. : . ✓ The quotient is .

Answer. Zeros — three positive, matching the first option in Step 2.

Common Mistakes to Avoid

• Counting sign changes in a polynomial that is out of order. The rule reads consecutive coefficients in descending powers. Put the polynomial in standard form first.
• Treating a missing term as a sign change. A zero coefficient is skipped. In the signs are : two changes, not more.
• Flipping the wrong terms in . Only odd powers change sign: but . The constant never changes.
• Reading the rule as an exact count. Three sign changes means or . Only a count of or is exact.
• Forgetting that is neither positive nor negative. Factor out any power of before applying the rule, and remember to add those zeros back at the end.
• Expecting the rule to find the zeros. It counts; the rational root theorem and synthetic division find. Use them together.

Practice Problems

Work each one before opening the answer.

Problem 1. How many sign changes does have, and what are the possible numbers of positive real zeros?

Problem 2. Find for and state the possible numbers of negative real zeros.

Problem 3. Build the full table of possibilities for .

Problem 4. Use Descartes’ rule of signs to show that has no real zeros.

Problem 5. has rational-root candidates . Which of them does Descartes’ rule allow?

Problem 6. Apply Descartes’ rule of signs to .

Descartes’ rule of signs is a counting tool, and it earns its place by what it removes: candidates you never have to test, and cases you can close without solving. It sits naturally between the rational root theorem, which supplies the candidates, and the complex zeros lesson, which explains where the missing real zeros went.