Chapter 4, Problem 89RE

To determine

Design the polynomial function with the following characteristics:

Degree $6$ ;four real zeros, one of multiplicity $3$ ; $y-$ intercept $3$ ;behaves like $y=-5x^6$ for large values of $\left|x\right|$ .

Is this polynomial unique?

Answer to Problem 89RE

The polynomial function is $-5x^6+b{x}^5+c{x}^4+d{x}^3+e{x}^2+fx+3=y$ and all terms are explained.

Explanation of Solution

Given:

Characteristics: Degree $6$ ;four real zeros, one of multiplicity $3$ ; $y-$ intercept $3$ ;behaves like $y=-5x^6$ for large values of $\left|x\right|$ .

Calculation:

The polynomial should have a degree of $6$ with $4$ real distinct roots out of which $1$ will have a multiplicity of $3$ . Further it has a $y-$ intercept of $3$ and it behaves as $y=-5x^6$ for large values of $x$ .

In accordance with the function the possible polynomial function can be,

  $-5x^6+b{x}^5+c{x}^4+d{x}^3+e{x}^2+fx+3=y$

Considering the roots of the equation to be $m,n,o,p,p,p$ . So,the equation is $-5\left(x-m\right)\left(x-n\right)\left(x-o\right)\left(x-p\right)\left(x-p\right)\left(x-p\right)+3=y$

Where,

  $m\times n\times o\times p^3=3$

The condition that will be same for everyone will be that the behavior of the function for large values. The $y-$ intercept will be $3$ for everyone.

Further adding some more characteristics such as symmetry and naming real zeroes will help in determining the exact polynomial.

The symmetry will be determining the repeated roots of the function and real zeroes will help to know the points at which the graph of the fun function touches the real axis.

Conclusion:

Hence,thepolynomial function is $-5x^6+b{x}^5+c{x}^4+d{x}^3+e{x}^2+fx+3=y$ and all terms are explained.