Chapter 4.6, Problem 52E

(a)

To determine

To find: The graph the given equations and the zeros and the coefficient of polynomials.

Answer to Problem 52E

The required equation is $x=\frac{1}{4}{\left(y+1\right)}^2+1$ .

Explanation of Solution

Given equation:

  $f\left(x\right)=x^2-5x+6$

  $g\left(x\right)=x^3-7x+6$

  $h\left(x\right)=x^4+2x^3+x^2+8x-12$

  $k\left(x\right)=x^5-3x^4-9x^3+25x^2-6x$

Consider the equation:

  $f\left(x\right)=x^2-5x+6$

Graph the above equation:

  

Consider the equation:

  $g\left(x\right)=x^3-7x+6$

Graph the above equation:

  

Consider the equation:

  $h\left(x\right)=x^4+2x^3+x^2+8x-12$

Graph the above equation:

  

Consider the equation:

  $k\left(x\right)=x^5-3x^4-9x^3+25x^2-6x$

Graph the above equation:

  

It is seen from the above graphs formed by the four equations that the sum of the roots are the negative coefficient of ${\left(x\right)}^{n-1}$

Hence, the required sum of the roots of the equation is ${\left(x\right)}^{n-1}$ .

(b)

To determine

To find: The graph the given equations and the product of the roots.

Answer to Problem 52E

The required equation is $x=\frac{1}{4}{\left(y+1\right)}^2+1$ .

Explanation of Solution

Given equation:

  $f\left(x\right)=x^2-5x+6$

  $g\left(x\right)=x^3-7x+6$

  $h\left(x\right)=x^4+2x^3+x^2+8x-12$

  $k\left(x\right)=x^5-3x^4-9x^3+25x^2-6x$

Consider the equation:

  $f\left(x\right)=x^2-5x+6$

Graph the above equation:

  

Consider the equation:

  $g\left(x\right)=x^3-7x+6$

Graph the above equation:

  

Consider the equation:

  $h\left(x\right)=x^4+2x^3+x^2+8x-12$

Graph the above equation:

  

Consider the equation:

  $k\left(x\right)=x^5-3x^4-9x^3+25x^2-6x$

Graph the above equation:

  

It is seen from the above graphs formed by the four equations that the product of the roots are ${\left(-1\right)}^n$ times the constant term.

Hence, the required product of the roots of the equation is ${\left(-1\right)}^n$ the constant term.