Worked example 1
Solve a cubic equation
Solve x³ − 2x² − 5x + 6 = 0, given that x = 1 is a root.
- Divide by (x − 1).
- The quotient is x² − x − 6.
- Factorise the quotient as (x − 3)(x + 2)
Answer: x = −2, 1 or 3
Higher Mathematics
Higher Maths Cubic and quartic equations
Practise cubic and quartic equations for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Factorise and solve cubic or quartic polynomial equations using known or discoverable factors.
Before you start
- Use the factor theorem.
- Divide a polynomial by a linear factor.
- Solve quadratic equations exactly.
Explanation
A Higher polynomial equation is usually reduced after one linear factor is found or given. Polynomial division leaves a quadratic or cubic factor that can be solved further.
A quartic may factor into two quadratics or become a quadratic in x²; only real roots are reported unless the question says otherwise.
Method and rules
- If f(a) = 0, then (x − a) is a factor.
- Solve every remaining factor equal to zero.
- For an even-power quartic, let u = x² when useful.
Worked examples
Worked example 2
Method check
Solve x⁴ − 5x² + 4 = 0.
- Factorise as a quadratic in x²
So: x = −2, −1, 1 or 2
Watch out
- Stopping after finding the given root.
- Losing a term during polynomial division.
- Treating x² = a as having only the positive root
Exam reminder
For cubic and quartic equations, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.