Chapter 4, Problem 50RE

(a)

To determine

To show if the polynomial function is even or odd.

Answer to Problem 50RE

The graph of the polynomial function is even.

Explanation of Solution

Given:

  

Concept Used:

The direction of motion in the same direction is positive

The direction of motion in opposite direction is negative.

Calculation:

The end behavior of polynomial function are in same direction.

Hence , the degree of polynomial is even.

Conclusion:

The graph of polynomial function has even degree.

(b)

To determine

If the leading coefficient is negative or positive.

Answer to Problem 50RE

The leading coefficient is positive.

Explanation of Solution

Given:

  

Concept Used:

The coordinate axis:

x -axis

Right side is positive.

Left side is negative

Calculation:

Since the graph is on the right side of the x -axis.

Hence, the leading coefficient is positive.

Conclusion:

The leading coefficient is positive.

(c)

To determine

If the function is even, odd or neither .

Answer to Problem 50RE

The function is even.

Explanation of Solution

Given:

  

Concept Used:

A graph is symmetric about y-axis if $(a,b)$ is on the graph then so is $(-a,b)$ .

Calculation:

The graph is symmetric to y -axis. Therefore, the function is even.

Conclusion:

The graph of polynomial function is even.

(d)

To determine

If polynomial has factor, then why $x^2$ is necessarily a factor?

Answer to Problem 50RE

The $x^2$ is the necessary factor for the polynomial function..

Explanation of Solution

Given:

  

Concept Used:

Equation of the parabola is

  $y=x^2$

Calculation:

Since the graph is likely as parabola at $x=0$ . This implies $x^2$ be the necessary factor.

Conclusion:

The $x^2$ is the necessary factor of the polynomial.

(e)

To determine

The minimum degree of the polynomial.

Answer to Problem 50RE

The minimum degree is 8.

Explanation of Solution

Given:

  

Concept Used:

Turning points: n -1

Degree: n

Calculation:

Since there are seven turning points in the graph.

So, the minimum degree of polynomial function is 8.

Conclusion:

The minimum degree is 8.

(f)

To determine

To formulate the different polynomial functions.

Answer to Problem 50RE

The polynomial function $f(x)=x^2(x+3)(x+2)(x+1)(x-1)(x-2)(x-3)$ .

Explanation of Solution

Given:

  

Calculation:

With the minimum degree of polynomial function 8. We have ,

  $f(x)=x^2(x+3)(x+2)(x+1)(x-1)(x-2)(x-3)$

Conclusion:

The polynomial $f(x)=x^2(x+3)(x+2)(x+1)(x-1)(x-2)(x-3)$ .