Equivalent expressions, factors and rational powers
Choose an equivalent form that reveals structure while preserving any restrictions.
What you will tackle
- Add, subtract and multiply polynomials
- Factor common factors, differences of squares and trinomials
- Simplify rational expressions while retaining excluded inputs
- Translate between radicals and rational exponents
Operations preserve a relationship, not a visual pattern
Distribute a negative sign across every term when subtracting polynomials. Factor a common factor first: 3x² − 12 = 3(x − 2)(x + 2). The identity a² − b² = (a − b)(a + b) applies to a difference of squares, not a sum.
A simplified rule retains the original domain
For x ≠ 3, (x² − 9)/(x − 3) = x + 3. Cancelling a factor does not define the original expression at x = 3. For positive a, a^(m/n) is the nth root of a^m. Real even roots require nonnegative radicands, and negative exponents also require a nonzero base.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Expand (x + 2)(x − 5), factor 2x² − 18, and simplify (x² − 9)/(x − 3) with its restriction.
Show a starting hint
Cancellation applies to factors, not separate added terms.
Compare with the explained answer
- Expansion gives x² − 3x − 10.
- Factor out 2, then use a difference of squares: 2(x − 3)(x + 3).
- Cancel x − 3 only for x ≠ 3, obtaining x + 3.
Answer / response criteria: x² − 3x − 10; 2(x − 3)(x + 3); x + 3 with x ≠ 3.
Guided 2
Evaluate 32^(2/5) and factor x³ + 2x² − 9x − 18 by grouping.
Compare with the explained answer
- The fifth root of 32 is 2, so its square is 4.
- Group x²(x + 2) − 9(x + 2).
- Factor (x + 2)(x² − 9) = (x + 2)(x − 3)(x + 3).
Answer / response criteria: 4; (x + 2)(x − 3)(x + 3).
Independent practice
Independent 1
Simplify (3x² + 2x − 1) − (x² − 4x + 5), factor x² + 5x + 6, and evaluate 27^(2/3).
Compare with the explained answer
- Subtract each term: 2x² + 6x − 6.
- The factors of 6 summing to 5 are 2 and 3, giving (x + 2)(x + 3).
- The cube root of 27 is 3; squaring gives 9.
Answer / response criteria: 2x² + 6x − 6; (x + 2)(x + 3); 9.
Independent 2
Evaluate 81^(3/4) and factor x³ − x² − 4x + 4. Verify the factorisation by regrouping.
Compare with the explained answer
- The fourth root of 81 is 3; cubing gives 27.
- Group x²(x − 1) − 4(x − 1).
- The result is (x − 1)(x − 2)(x + 2).
Answer / response criteria: 27; (x − 1)(x − 2)(x + 2).
Application practice
Application 1
A square garden has side x + 3 metres. A square patio of side x is removed. Find the planted area in a form that makes its rate of change clear, and state a physical restriction.
Compare with the explained answer
- Subtract areas: (x + 3)² − x².
- Expanding and cancelling x² leaves 6x + 9.
- The area increases by 6 m² per added metre of x; lengths require x ≥ 0.
Answer / response criteria: 6x + 9 m², for x ≥ 0.
Application 2
Two similar containers have volume ratio 64. Express their surface-area ratio using a rational power. A model has zero output when x³ + x² − 16x − 16 = 0; factor to locate its roots and retain only positive x for a length.
Compare with the explained answer
- The area ratio is 64^(2/3) = 16.
- Grouping gives (x + 1)(x² − 16) = (x + 1)(x − 4)(x + 4).
- The algebraic roots are −1,4,−4; only 4 is a positive length.
Answer / response criteria: Area ratio 16; roots −1,4,−4; physical positive root 4.
What can you explain now?
Two similar containers have volume ratio 64. Express their surface-area ratio using a rational power. A model has zero output when x³ + x² − 16x − 16 = 0; factor to locate its roots and retain only positive x for a length.
Compare with the explanation
Area ratio 16; roots −1,4,−4; physical positive root 4.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide