Question

The illustration shows the graph of a polynomial function.

(a) Is the degree of the polynomial even or odd?

(b) Is the leading coefficient positive or negative?

(c) Is the function even, odd, or neither?

(d) Why is $x^2$necessarily a factor of the polynomial?

(e) What is the minimum degree of the polynomial?

(f) Formulate five different polynomial functions whose

graphs could look like the one shown. Compare yours to those of other students. What similarities do you see?

What differences?

Short answer

(a) Even

(b) Positive

(c) Even

(d) The function shown in the graph has no y-intercept and also the function is even.

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

(e)$8$

Step-by-step solution

Part (a) Step 1. Degree of the polynomial

The degree of the graph will be even if the the graph is of polynomial.

Part (b) Step 1. Leading coefficient

The graph is upwards therefore the leading coefficient is positive.

part (c) Function even or odd.

Since the function in the graph is symmetric about $y$axis, therefore the function is even.

Part (d) Step 1. Reason for x2 necessarily a factor of the polynomial

The function shown in the graph has no y-intercept and also the function is even.

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

Part (e) Step 1. Minimum degree of polynomial

The minimum degree of the graph is$8$

Part (f) Step 1. General form of the function similar to the given in the graph

The general form of the polynomial function shown in the graph is:

$$\begin{gathered} a{x}^{2k}-b{x}^{2k-2}+c{x}^{2k-4}+.......+e{x}^2=y \\ k\ge 4 \end{gathered}$$