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
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide