Polynomials practice
Factor twelve expanded polynomials completely over the integers, mixing greatest common factors, quadratics, grouping, cube identities, substitution patterns, repeated factors and irreducible quadratic factors.
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Factor each polynomial completely over the integers.
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Start with the full numerical and variable common factor. Then use the polynomial’s term count and exponent pattern to choose quadratic factoring, grouping, a cube identity, difference of squares or an even-power substitution.
Inspect every resulting factor again. Repeated factors keep their multiplicity, and a primitive quadratic remains only when its discriminant is not a perfect square. Multiply the complete product to check the original coefficients.
Learning objective
Choose an efficient factoring method, extract the full content and continue until every remaining factor is primitive and irreducible over the rationals.
Worked example
2(x − 2)(x + 2)(x² + 4).
Extract 2, factor x⁴ − 16 as a difference of squares, then factor x² − 4. The quadratic x² + 4 is irreducible over the rationals.
Teaching notes
Use quadratic and cubic degrees for a shorter decision chain, then degrees four through six for repeated patterns and even-power substitutions. Degree alone does not identify the technique: remove the numerical and variable greatest common factor before inspecting term count, exponents and signs. The mixed preset combines both degree groups.
Complete means every remaining nonconstant factor is irreducible over the rationals. For x⁴ − 5x² + 4, let u = x² to get (u − 1)(u − 4). Substitute back and continue: (x − 1)(x + 1)(x − 2)(x + 2). By contrast, x² + 4 stays quadratic when factoring over the integers.
No. This is mixed-technique practice for choosing a method and carrying it to completion. Use the related cube or grouping activities below when a learner needs a focused introduction to one method.
The editable studio selects the polynomial degree group for twelve questions. Its download and worked key follow that selection. The ready-made files above retain the default mixed-degree edition; their contents do not change with a query link.
Multiply every factor, including the numerical content and repeated factors, to recover the original polynomial. A correct first step can still be incomplete, so inspect each factor again before comparing with the key.
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