Chapter 5.6, Problem 36HP

To determine

To write: a polynomial function that has double root of 1 and a double root of $-5$ and graph the function.

Answer to Problem 36HP

The function is $f(x)={\left(x-1\right)}^2{\left(x+5\right)}^2$ .

  

Explanation of Solution

Given information:

The function has double root of 1 and a double root of $-5$ .

Formula used:

Factor Theorem: If $f(x)$ is a polynomial of degree $n\ge 1$ and a is any real number , then $(x-a)$ is a factor of $f(x)$ , if $f(a)=0$ .

Remainder Theorem:- If a polynomial $P(x)$ is divided by $x-r$ , the remainder s a constant $P(r)$ , and

  $P(x)=Q(x)(x-r)+P(r)$ ,

Where $Q(x)$ is a polynomial with degree one less than $P(x)$ .

Calculation:

Consider ,

The function $f(x)$ .

Since, the function has double root of 1 and a double root of $-5$ .

Using factor theorem we get $(x-1)\text{ and }(x+5)$ are the factor of the polynomial.

Therefore we get,

  $f(x)={\left(x-1\right)}^2{\left(x+5\right)}^2$.

Now , the graph of the function is below. Both factors are family of parabolic.

  

Figure 1.

Therefore, the polynomial function is $f(x)={\left(x-1\right)}^2{\left(x+5\right)}^2.$