Chapter 4, Problem 12PS

(a)

To determine

To calculate: The zeroes, the sum of zeroes and the product of zeroes for the function given in the table below,

$\begin{array}{llll}\text{Function}\hfill & \text{Zeros}\hfill & \text{Sum of Zeros}\hfill & \text{Product of Zeros}\hfill \\ f_1\left(x\right)=x^2-5x+6\hfill & \hfill & \hfill & \hfill \\ f_2\left(x\right)=x^3-7x+6\hfill & \hfill & \hfill & \hfill \\ f_3\left(x\right)=x^4+2x^3+x^2+8x-12\hfill & \hfill & \hfill & \hfill \\ f_4\left(x\right)=x^5-3x^4-9x^3+25x^2-6x\hfill & \hfill & \hfill & \hfill \end{array}$

(b)

To determine

The conjecture for the sum of zeros for the polynomial function with the help of the below table:

$\begin{array}{llll}\text{Function}\hfill & \text{Zeros}\hfill & \text{Sum of Zeros}\hfill & \text{Product of Zeros}\hfill \\ f_1\left(x\right)=x^2-5x+6\hfill & 2,3\hfill & 5\hfill & 6\hfill \\ f_2\left(x\right)=x^3-7x+6\hfill & -3,1,2\hfill & 0\hfill & -6\hfill \\ f_3\left(x\right)=x^4+2x^3+x^2+8x-12\hfill & -3,1,\pm 2i\hfill & -2\hfill & -12\hfill \\ f_4\left(x\right)=x^5-3x^4-9x^3+25x^2-6x\hfill & -3,0,2,2\pm \sqrt{3}\hfill & 3\hfill & 0\hfill \end{array}$

(c)

To determine

The conjecture for the product of zeros for the polynomial function with the help of the below table:

$\begin{array}{llll}\text{Function}\hfill & \text{Zeros}\hfill & \text{Sum of Zeros}\hfill & \text{Product of Zeros}\hfill \\ f_1\left(x\right)=x^2-5x+6\hfill & 2,3\hfill & 5\hfill & 6\hfill \\ f_2\left(x\right)=x^3-7x+6\hfill & -3,1,2\hfill & 0\hfill & -6\hfill \\ f_3\left(x\right)=x^4+2x^3+x^2+8x-12\hfill & -3,1,\pm 2i\hfill & -2\hfill & -12\hfill \\ f_4\left(x\right)=x^5-3x^4-9x^3+25x^2-6x\hfill & -3,0,2,2\pm \sqrt{3}\hfill & 3\hfill & 0\hfill \end{array}$