Chapter 4.1, Problem 120AYU

(a)

To determine

To find: whether the degree of the polynomial even or odd

Answer to Problem 120AYU

The degree of this polynomial is even

Explanation of Solution

Given:

  

Calculation:

Now let us analyze this graph with respect to the problem statement.

This is a characteristic graph of a polynomial function. The degree of this polynomial is even or else the graph will not be that of a polynomial function.

Conclusion:

The degree of this polynomial is even

(b)

To determine

To find: whether the leading coefficient positive or negative

Answer to Problem 120AYU

The leading coefficient is positive,

Explanation of Solution

Calculation:

  

The leading coefficient is positive, as the graph is upwards.

Conclusion:

The leading coefficient is positive,

(c)

To determine

To find: whether the function even, odd, or neither

Answer to Problem 120AYU

The given function is even

Explanation of Solution

Calculation:

The polynomial function shown in the graph will be even as it is symmetric about y-axis.

Even functions are symmetric about y-axis.

Conclusion:

The given function is even

(d)

To determine

To explain: why $x^2$ is a factor of the polynomial

Answer to Problem 120AYU

The polynomial function shown in the graph is even, has no y-intercept.

Explanation of Solution

Calculation:

The polynomial function shown in the graph is even, has no y-intercept.

Therefore, $x^2$ can be taken common from the whole function

Hence, it can have $x^2$ as a factor.

Conclusion:

The polynomial function shown in the graph is even, has no y-intercept.

(e)

To determine

To find: What is the minimum degree of the polynomial?

Explanation of Solution

Calculation:

The minimum degree of the graph should be 8. Though the crossing points are only 6 with one touching point but with respect to other conditions, the only possibility is a polynomial with degree 8.

Conclusion:

(f)

To determine

To find: Formulate five different polynomials whose graphs could look like the one shown. Compare yours to those of other students. What similarities do you see? What differences?

Explanation of Solution

Calculation:

The graph shown can be any graph for some condition such as the degree of the polynomial should be greater than 8 and should be even. Also the leading coefficient should be positive and $x^2$ can be taken out of those polynomials and with alternating positive negative sign. Therefore, the general equation of such a polynomial will be

  $a{x}^{2k}-b{x}^{2k-2}+c{x}^{2k-4}+\mathrm{...}+e{x}^2=yk\ge 4$

Conclusion: