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Test: Higher-degree rational equations

This test focuses on cubic, biquadratic and other higher-degree equations. Students factor expressions, use substitution and verify which candidate values satisfy the original equation.

Question qty: 10
Duration in minutes: 18
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Sample questions

  • The intersection equation of a line and parabola has discriminant d. How many real intersection points are there?
  • How do algebraic and graphical equation solving differ?
  • Solve x³ = v.
  • A rectangle has sides x and x − a, and area b. Find the positive x.
  • What restriction applies to a rational equation with denominator x − (a)?
  • Solve (x − (a))(x − (b)) = 0.
  • Solve x⁴ = v.
  • Solve x⁴ − sx² + p = 0.

Higher-degree rational equations

A higher-degree equation can often be simplified by factorisation, grouping or a substitution such as t = x². For a rational equation, first identify every value excluded by its denominators.

Finding roots

After finding one polynomial root, factor out x − a and solve the remaining lower-degree equation. Reverse every substitution to recover all original variable values. Check every candidate in the original equation because transformations may introduce inadmissible solutions.

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