Edexcel IAL - Further Pure Mathematics 1- 1.5 Conjugate Roots and Real Roots of Cubics- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 1- 1.5 Conjugate Roots and Real Roots of Cubics -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 1- 1.5 Conjugate Roots and Real Roots of Cubics -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 1.5 Conjugate Roots and Real Roots of Cubics
Conjugate Complex Roots and a Real Root of a Cubic Equation
Consider a cubic polynomial with real (or integer) coefficients:
\( f(x) = ax^3 + bx^2 + cx + d \)
If this polynomial has a complex root, then its conjugate is also a root.
If \( z \) is a root, then \( \overline{z} \) is also a root.
This is called the Complex Conjugate Root Theorem.
Since a cubic has three roots, if two of them are complex conjugates, the third root must be real.
Method
- Use a given complex root to write two factors.
- Multiply to form a quadratic factor.
- Divide the cubic by this quadratic factor.
- The remaining factor gives the real root.
Forming the Quadratic Factor
If a root is \( a + bi \), the other is \( a – bi \).
The corresponding quadratic factor is:
\( (x – (a + bi))(x – (a – bi)) = (x – a)^2 + b^2 \)
Example
The polynomial \( f(x) = x^3 – 4x^2 + 5x – 2 \) has a root \( 1 + i \). Find the other complex root and the real root.
▶️ Answer / Explanation
The conjugate root is \( 1 – i \).
Form the quadratic factor:
\( (x – (1 + i))(x – (1 – i)) = (x – 1)^2 + 1 = x^2 – 2x + 2 \)
Divide \( f(x) \) by \( x^2 – 2x + 2 \).
\( x^3 – 4x^2 + 5x – 2 = (x^2 – 2x + 2)(x – 2) \)
The real root is \( x = 2 \).
Example
The polynomial \( f(x) = x^3 + 3x^2 + 5x + 3 \) has a root \( -1 + i \). Find all its roots.
▶️ Answer / Explanation
The conjugate root is \( -1 – i \).
Quadratic factor:
\( (x + 1)^2 + 1 = x^2 + 2x + 2 \)
Divide:
\( x^3 + 3x^2 + 5x + 3 = (x^2 + 2x + 2)(x + 1) \)
The real root is \( x = -1 \).
So the roots are:
\( -1 + i,\ -1 – i,\ -1 \)
Example
The cubic polynomial \( f(x) = x^3 – 5x^2 + 9x – 13 \) has a root \( 2 + 3i \). Find the other two roots.
▶️ Answer / Explanation
The conjugate root is \( 2 – 3i \).
Quadratic factor:
\( (x – 2)^2 + 9 = x^2 – 4x + 13 \)
Divide:
\( x^3 – 5x^2 + 9x – 13 = (x^2 – 4x + 13)(x – 1) \)
The real root is \( x = 1 \).
So the roots are:
\( 2 + 3i,\ 2 – 3i,\ 1 \)