Conjugate roots and rational coefficients
For a quadratic with rational coefficients and nonzero leading coefficient, let an irrational root be p+q√r, where p and q are rational, q≠0, and r is a positive nonsquare integer. Its other root is p−q√r. Their sum and product are rational because the surd terms cancel
- Keep the rational part The conjugate of p+q√r keeps p and changes only the sign of the surd term \(\displaystyle 2+\sqrt{3}\Rightarrow2-\sqrt{3}\)
- Use x minus each root A root r gives a factor x-r in the polynomial \(\displaystyle r=2+\sqrt{3}\Rightarrow x-r=x-\left(2+\sqrt{3}\right)\)
- Multiply as a difference of squares Conjugate factors cancel their surd terms when they are multiplied \(\displaystyle \left(x-\left(2+\sqrt{3}\right)\right)\left(x-\left(2-\sqrt{3}\right)\right)=(x-2)^2-3\)
- Collect rational coefficients After the surds cancel, expand the remaining rational expression and apply any leading scale \(\displaystyle 2\left[(x-2)^2-3\right]=2x^2-8x+2\)
Worked example
One root of a quadratic with rational coefficients is α = 1 + 2√3. The leading coefficient is 2. Construct the quadratic equation
- \(\displaystyle \beta=1-2\sqrt{3}\)Keep the rational part and reverse only the sign of the surd term
- \(\displaystyle \alpha+\beta=(1+2\sqrt{3})+(1-2\sqrt{3})=2\)Add the conjugate roots so the radical terms cancel
- \(\displaystyle \alpha\beta=(1+2\sqrt{3})(1-2\sqrt{3})=1-(2\sqrt{3})^2=1-12=-11\)Multiply the conjugates as a difference of squares
- \(\displaystyle 2\left(x-\alpha\right)\left(x-\beta\right)=2\left[(x-1)^2-12\right]=2x^2-4x-22=0\)Form the two root factors, apply the leading coefficient and expand the rational result
Common mistakes
Difficulty levels
Questions about this topic
What is the conjugate of p + q√r?+
For rational p and q with q≠0 and a positive nonsquare integer r, p+q√r is irrational and its conjugate is p−q√r. Keep p and change only the surd sign. The rational-coefficient condition is needed to conclude that this is the other quadratic root
Why do conjugate roots give rational coefficients?+
Their sum and product are rational because the surd terms cancel. With leading coefficient a, expanding the factors gives ax² − a(root sum)x + a(root product), so the coefficients are rational
Why is the factor for a root r written as x - r?+
Substituting x = r into x - r gives zero, so x - r is the factor associated with the root r
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 3: Quadratic functions