Chapter 6, Problem 1TE

For Exercises 1–4, (a) give the degree of the polynomial and (b) give the leading coefficient.

1. $3x^4-5x^2+\frac{2}{3}$

To determine

To give: The degree of the polynomial $3x^4-5x^2+\frac{2}{3}$.

Answer to Problem 1TE

The polynomial $3x^4-5x^2+\frac{2}{3}$ is a fourth degree polynomial.

Explanation of Solution

Formula used:

General form of polynomial:

“1. The general form of the polynomial is given by $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$, where $a_0$ and each coefficient of x are real numbers and each power of x is positive integer.

2. If $a_n\ne 0$, n is the highest power of x, $a_n$ is called the leading coefficients and n is the degree of the polynomial”.

Calculation:

The given polynomial $3x^4-5x^2+\frac{2}{3}$ which is in the form of $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$, where n is the degree of the polynomial.

The polynomial $3x^4-5x^2+\frac{2}{3}$ has the highest power term as $3x^4$ and is of degree 4.

Therefore, the polynomial $3x^4-5x^2+\frac{2}{3}$ is a fourth degree polynomial.

To determine

To give: The leading coefficient of the polynomial $3x^4-5x^2+\frac{2}{3}$.

Answer to Problem 1TE

The leading coefficient of the polynomial $3x^4-5x^2+\frac{2}{3}$ is 3.

Explanation of Solution

From the polynomial $3x^4-5x^2+\frac{2}{3}$, it is noted that the highest degree term is $3x^4$ and its coefficient is 3.

By the above definition (2), it is known that if $a_n\ne 0$, n is the highest power of x and $a_n$ is called the leading coefficients.

Therefore, the leading coefficient is 3 which is the coefficient of highest degree term.