Edexcel IAL - Further Pure Mathematics 1- 1.6 Conjugate and Real Roots of Quartic Equations- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 1- 1.6 Conjugate and Real Roots of Quartic Equations -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 1- 1.6 Conjugate and Real Roots of Quartic Equations -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 1.6 Conjugate and Real Roots of Quartic Equations
Quartic Equations with Complex and Real Roots
A quartic equation has the form:
\( f(x) = ax^4 + bx^3 + cx^2 + dx + e \)
When the coefficients are real, any complex roots occur in conjugate pairs. If \( z \) is a root, then \( \overline{z} \) is also a root.
This allows quartic equations to be factorised into quadratic factors.
Key Ideas
- Complex roots occur in conjugate pairs.
- Each conjugate pair forms a quadratic factor.
- Known real roots give linear factors.
- After removing known factors, the remaining quadratic can be solved.
Example
\( f(x) = x^4 – x^3 – 5x^2 + 7x + 10 \) Given that \( x = 2 + i \) is a root, find the three other roots.
▶️ Answer / Explanation
Since coefficients are real, \( 2 – i \) is also a root.
Form the quadratic factor:
\( (x – (2 + i))(x – (2 – i)) = (x – 2)^2 + 1 = x^2 – 4x + 5 \)
Divide \( f(x) \) by \( x^2 – 4x + 5 \).
\( f(x) = (x^2 – 4x + 5)(x^2 + 3x + 2) \)
Factorise the remaining quadratic:
\( x^2 + 3x + 2 = (x + 1)(x + 2) \)
So the roots are:
\( x = 2 + i,\ 2 – i,\ -1,\ -2 \)
Example
\( g(x) = x^4 – x^3 + 6x^2 + 14x – 20 \) Given that \( g(1) = 0 \) and \( g(-2) = 0 \), solve \( g(x) = 0 \) completely.
▶️ Answer / Explanation
Since \( g(1) = 0 \), \( x – 1 \) is a factor.
Since \( g(-2) = 0 \), \( x + 2 \) is a factor.
So:
\( g(x) = (x – 1)(x + 2)(x^2 + ax + b) \)
Expanding the known factors:
\( (x – 1)(x + 2) = x^2 + x – 2 \)
So:
\( g(x) = (x^2 + x – 2)(x^2 + ax + b) \)
Comparing with the original polynomial gives:
\( a = -2,\ b = 10 \)
So:
\( g(x) = (x – 1)(x + 2)(x^2 – 2x + 10) \)
Solve \( x^2 – 2x + 10 = 0 \):
\( x = \dfrac{2 \pm \sqrt{4 – 40}}{2} = 1 \pm 3i \)
So the four roots are:
\( x = 1,\ -2,\ 1 + 3i,\ 1 – 3i \)