Problem 4 - College Algebra

Question

Determine which functions are polynomial functions. For those that are, identify the degree. $$ g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x $$

Step-by-Step Solution

Answer

Yes, the function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\) is a polynomial function and its degree is 7.

Step 1: Identification of a Polynomial Function

To determine whether this function is a polynomial, each of its terms must comply with the form of \(a_nx^n\), where \(a_n\) is a constant and \(n\) is a non-negative integer. In \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\), all terms comply with this form; thus, it is a polynomial function.

Step 2: Identification of the Degree of the Polynomial

The degree of a polynomial is the highest power in the polynomial. In this polynomial, the highest power is 7 on the x variable in the term \(6 x^{7}\), so the degree of this polynomial is 7.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is essential for classifying and solving polynomial equations. In simple terms, the degree of a polynomial is the highest power of the variable (usually 'x') that appears in the polynomial's expression. For example, take a polynomial in the form of \( ax^n + bx^{n-1} + \dots + px + q \), where \( a, b, \dots, p \), and \( q \) are coefficients, which can be any number. The degree of this polynomial would be \( n \), assuming \( a \) is not zero.

When looking at the degree, only consider the highest power of 'x' that has a non-zero coefficient. This power tells us much about the polynomial, including the maximum number of solutions and the shape of its graph. For example, a second-degree polynomial forms a parabola, while higher degrees can indicate more complex curves. Remember that only whole numbers (non-negative integers) are considered for the degrees of the polynomial terms.

Polynomial Function Identification

Identifying a polynomial function is the first step in analyzing it. A function is considered a polynomial function if it can be expressed in the standard polynomial form, which is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \) where the coefficients \( a_n, a_{n-1}, \dots, a_1, a_0 \) are real numbers and \( n \) is a non-negative integer. Additionally, the powers of 'x' must be whole numbers.

To verify a given function is a polynomial, check each term to ensure it fits the form of \( a_nx^n \) mentioned earlier. If there's a term with a variable in the denominator or with a negative or fractional exponent, it's not a polynomial. In the example of \( g(x)=6 x^{7}+\pi x^{5}+ \frac{2}{3} x \), all terms meet the requirements since all coefficients are real numbers, and the exponents are non-negative integers.

Polynomial Terms

Polynomials consist of one or more terms added together, where each polynomial term is a product of a coefficient and the variable raised to a non-negative integer power. The general form of a polynomial term is \( ax^n \) where \( a \) is the coefficient, which can be any real number, and \( n \) is the exponent, which must be a non-negative integer.

Examples of Polynomial Terms

• \(3x^2\) is a polynomial term where \(3\) is the coefficient and \(2\) is the exponent.
• \(-5x\) is a polynomial term where \( -5\) is the coefficient and \(1\) is the exponent (as \(x\) is equivalent to \(x^1\)).
• \(7\) is also a polynomial term, known as a constant term, where \(7\) is the coefficient, and the exponent of \(x\) is \(0\) since any number raised to the power of \(0\) equals \(1\).

Terms can be combined to form polynomials, such as \(3x^2 - 5x + 7\), and the combination of these terms' degrees gives us the polynomial's degree. Clear identification of individual terms helps us to understand polynomials' structure and prepare for operations like addition, subtraction, or simplification.