Polynomials practice

Descartes Rule of Signs worksheets

Use sign changes in P(x) and P(−x) to list every possible number of positive and negative real zeros for twelve degree-two through degree-six polynomials.

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List the possible numbers of positive and negative real zeros, counting multiplicity. These are possibilities, not the actual zeros.

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Count sign changes, then list every permitted count

Write P(x) in descending powers and count changes only between consecutive nonzero coefficients. Missing powers contribute no sign and must be skipped.

For P(−x), change the sign of every odd-power term and keep every even-power term unchanged. If V is the variation count, the permitted real-zero counts are V, V − 2, and so on to zero or one. These are possible counts including multiplicity, not the polynomial’s actual zeros.

Learning objective

What this activity builds

Count sign variations correctly through missing powers and convert each variation count into its complete descending-by-two possibility list.

Worked example

For P(x) = x⁴ − 3x³ + 2x − 5, list the possible positive and negative real-zero counts.

Positive: V(P(x)) = 3, so 3 or 1. Negative: V(P(−x)) = 1, so 1.

Ignore the missing x² term when counting. To form P(−x), change signs only on terms with odd exponents.

Teaching notes

Make the practice count

Order nonzero terms by descending exponent before counting changes.
Skip zero coefficients; they do not create or interrupt a sign change.
For P(−x), negate coefficients on odd powers and retain coefficients on even powers.
From V, list V, V − 2, and so on until reaching zero or one.
The possible counts include multiplicities and do not determine the polynomial’s actual positive or negative zeros.

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Choose degrees 2 through 6 in any combination. Keys show V(P(x)), V(P(−x)), and every possible positive and negative real-zero count.

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