Examples
The big rules! (Quadratic)
- \(\alpha+\beta=-\frac{b}{a}\)
- \(\alpha\beta=\frac{c}{a}\)
The big rules! (Cubic)
- \(\alpha+\beta+\gamma=-\frac{b}{a}\)
- \(\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}\)
- \(\alpha\beta\gamma=-\frac{d}{a}\)
The big rules! (Quartic)
- \(\alpha+\beta+\gamma+\delta=-\frac{b}{a}\)
- \(\alpha\beta+\alpha\gamma+\beta\gamma+\alpha\delta+\beta\delta+\gamma\delta=\frac{c}{a}\)
- \(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta=-\frac{d}{a}\)
- \(\alpha\beta\gamma\delta=\frac{e}{a}\)
The roots of a cubic equation \(ax^3+bx^2+cx+d=0\) are \(\alpha=1-2i\), \(\beta=1+2i\) and \(\gamma=2\). Find integer values for a, b, c and d.
- Finding the first equation: \(1-2i+1+2i+2=-\frac{b}{a}\)
- \(4=-\frac{b}{a}\)
- \(4a=-b\)
- Finding the second equation: \((1-2i)(1+2i)+2(1-2i)+2(1+2i)=\frac{c}{a}\)
- \(5+2-4i+2+4i=\frac{c}{a}\)
- \(9=\frac{c}{a}\)
- \(9a=c\)
- Find the third equation: \(2(1-2i)(1+2i)=-\frac{d}{a}\)
- \(10a=-d\)
- Just let a = 1
From this we can now find an equation: \(x^3-4x^2+9x-10=0\)
Exercise 4B Q1
Equation: \(2x^3+5x^2-2x+3=0\)
a. \(\alpha+\beta+\gamma=-\frac{5}{2}\)
b. \(\alpha\beta\gamma=-\frac{3}{2}\)
c. \(\alpha\beta+\beta\gamma+\gamma\alpha=-\frac{2}{2}=-1\)
d. \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=-1 / -\frac{3}{2}=\frac{2}{3}\)
Exercise 4C Q8 - The quartic equation \(x^4-16x^3+86x^2-176x+105=0\) has roots \(\alpha\), \(\alpha+k\), \(\alpha+2k\) and \(\alpha+3k\) for some real constant \(k\). Solve the equation.
- \(\alpha+\beta+\gamma+\delta=-\frac{b}{a}\)
- \(4\alpha+6k=16\)
- \(\alpha\beta\gamma\delta=\frac{e}{a}\)
Unfinished