Chapter 4.1, Problem 7E

To determine

Whether the given function is polynomial function or not. Also write the polynomial function is standard form and write the type, degree, and leading coefficient of the polynomial function.

Answer to Problem 7E

The given function is polynomial function.

Standard form is $h\left(x\right)=-\sqrt{7}{x}^4+8x^3+\frac{5}{3}{x}^2+x-\frac{1}{2}$ .

Degree of the polynomial function is 4.

Leading coefficient will be $-\sqrt{7}$ .

Given function is a quartic polynomial.

Explanation of Solution

Given information:

A function is given as

  $h\left(x\right)=\frac{5}{3}{x}^2-\sqrt{7}{x}^4+8x^3-\frac{1}{2}+x$

Concept used:

A polynomial function is of the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$

Terms of a polynomial function should be arranged in descending order according to its degree to express it in a standard form and degree of each term should be a positive integer or whole number. The coefficients should be real numbers.

Leading coefficient of a polynomial function is the coefficient of the leading term.

Degree of the polynomial is the degree of leading term or the height degree in the polynomial function.

For the polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$ the leading coefficient is $a_n$ and the degree is $n$ .

Calculation:

Consider the given function.

  $h\left(x\right)=\frac{5}{3}{x}^2-\sqrt{7}{x}^4+8x^3-\frac{1}{2}+x$

Now, degree of each term is a whole number and all coefficients are real.

So, the function is polynomial function.

The polynomial function can be written in its standard form by arranging the terms in descending order according to its degree.

Now, write the function in its standard form as shown:

  $h\left(x\right)=-\sqrt{7}{x}^4+8x^3+\frac{5}{3}{x}^2+x-\frac{1}{2}$

Now, degree of the polynomial function is 4.

Leading term is $-\sqrt{7}{x}^4$ .

So, leading coefficient will be $-\sqrt{7}$ .

As, degree of the polynomial function is 4, so it is a quartic polynomial.