Polynomials
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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Binomial Theorem worksheets
Find exact coefficients, identify requested ordinal terms, and write complete ordered expansions for twelve one-variable, two-variable, linear, higher-power, and hybrid binomials.
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Find requested coefficients and terms, then expand each binomial in descending powers of the first operand.
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Keep the printed operands in order. The (k + 1)th term is C(n,k) times the first operand to n − k times the second operand to k.
Ordinal terms follow descending powers of the first operand, even when the second variable has higher total degree. A complete expansion has n + 1 exact signed terms.
Learning objective
What this activity builds
Use T_(k+1) = C(n,k) first^(n−k) second^k while preserving the ordered operands and every signed term.
Worked example
Find the third term of (2x − y²)⁴.
24x²y⁴.
The third term uses k = 2: C(4,2)(2x)²(−y²)² = 6 · 4x² · y⁴.
Teaching notes
Make the practice count
- Keep the first and second operands in their printed order; ordinal terms descend in powers of the first operand.
- The (k + 1)th term uses k in the formula.
- Carry the second operand’s sign through its kth power.
- For a requested coefficient, match the complete target monomial before reading its signed numerical coefficient.
- A full expansion contains exactly n + 1 ordered terms.
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Choose one or two variables and linear or higher powers. Each sheet asks for four coefficients, four ordinal terms, and four complete expansions.
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Multiplying binomials worksheets
Expand ten products of two binomials. Choose among four expression structures, form every term pair and collect like terms with a separate worked key.
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