b) To find the roots of the equation \( f(x)=x^{3}-2 x^{2}-9 \), we start by solving the equation for \(x\), where \(x\) represents any real or complex number that satisfies the equation.
First, we set the equation to zero and find the values for \(x\):
\[ x^{3}-2 x^{2}-9 = 0 \]
This equation is a cubic equation. A cubic eqns can have 3 roots.
Upon solving this using various methods including factoring and the cubic formula, we find that the roots of the cubic equation are: \(3\), \(-1/2 - sqrt(11)*I/2\) and \(-1/2 + sqrt(11)*I/2\).
This means that when \(x\) is equal to these three values, the original equation is satisfied.
c) In order to draw the roots of this equation on an Argand diagram, we first notice that two of the roots are complex numbers. The Argand plane is a two-dimensional graphical representation of the complex plane where the x-axis represents the real part of a number, and the y-axis represents the imaginary part.
Since each root is expressed in the form of \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part, we can plot these roots onto an Argand diagram. In this case, the real part of the roots is the number before the plus or minus sign and the imaginary part is the number after the plus or minus sign (ignoring the \(i\), which indicates that it's an imaginary number).
Hence, the three roots expressed as complex numbers (including the real root) are: \(3+0j\), \(-0.5-1.6583123951777j\), and \(-0.5+1.6583123951777j\).
Each root is plotted as a point onto the Argand diagram. The coordinates for each point are given by the real and imaginary parts of each root, with \(i\) being ignored for the plot. The end result is three points on the Argand diagram representing the roots of the given equation.