Study notes
Surds & Polynomials
Simplifying and rationalising surds (including conjugate-pair denominators), solving simple surd equations, plus polynomial long division, the remainder and factor theorems, solving cubic/quartic equations, and partial fractions with distinct linear factors.
What this topic covers
- \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\): simplify a surd by extracting its largest square factor
- Rationalising a single-surd denominator: multiply top and bottom by that surd
- A conjugate pair removes a two-term surd denominator via the difference of two squares
- Solving a surd equation: isolate the surd, square both sides, then CHECK for extraneous roots
- Polynomial long division works just like numerical long division
- The remainder theorem finds a remainder WITHOUT doing the division
- Solve a cubic or quartic by finding ONE factor first, then dividing down to a quadratic
- Partial fractions with distinct linear factors split one fraction into simpler pieces
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