Problem 4 - College Algebra

Question

In Exercises \(1-10\), 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

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: Determine if the function is a polynomial

From the definition of a polynomial, we see that our function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\) involves only addition, multiplication and non-negative integer exponents of x, which are 7, 5, and 1. There aren't any variables under a square root, in the denominator, or inside a trigonometric or logarithmic function. We don't see that any coefficients are complex numbers, all of them are real numbers in this case. Thus, we can confirm the function g(x) is a polynomial function.

Step 2: Identify the degree of the polynomial

The degree of a polynomial function is the highest power of the variable (x) in the polynomial. Looking at the polynomial function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\), the highest power of x is 7. Thus, the degree of the polynomial is 7.

Key Concepts

degree of a polynomial

The degree of a polynomial is a critical aspect of understanding polynomial functions. It can give us insight into the function's behavior and characteristics as graphing and solving it becomes involved. The degree of a polynomial is defined as the highest power of the variable present when the polynomial is expressed in its standard form.
\[p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0\]
In this equation, the degree of the polynomial is \(n\) because \(n\) is the highest exponent on the variable \(x\).

• For instance, in the polynomial, \(g(x) = 6x^7 + \pi x^5 + \frac{2}{3}x\), the highest power of \(x\) is 7, making the degree of this polynomial 7.
• The degree tells us the number of roots or solutions the polynomial can possibly have.
• The degree influences the end behavior of the polynomial function graph.

Being able to identify the degree of the polynomial is foundational to analyzing polynomials in mathematics.

real coefficients

Coefficients are the numbers in front of the variables in a polynomial, and when we discuss real coefficients, we mean these numbers belong to the set of real numbers. Real numbers include all rational and irrational numbers, encompassing integers, fractions, and non-fractional numbers with decimal expansions that neither repeat nor terminate, like \(\pi\) or square roots of non-perfect squares.
When a polynomial has real coefficients, like \(g(x) = 6x^7 + \pi x^5 + \frac{2}{3}x\), we ensure that each coefficient, such as 6, \(\pi\), and \(\frac{2}{3}\), is a real number.

• Real coefficients guarantee that the polynomial expression is defined for all real numbers \(x\).
• They allow the polynomial to be graphed on the Cartesian plane smoothly.
• Complex coefficients, involving \(i\), are not present here, simplifying analysis and calculations.

Real coefficients keep the polynomial firmly within the realm of real and practical applications.

non-negative integer exponents

Within the definition of a polynomial, using non-negative integer exponents is essential. This means that the exponents on the variable \(x\) must be whole numbers starting from zero upwards: 0, 1, 2, and so on. Using exclusively non-negative integer exponents avoids the complications that come with fractions or negative numbers, which can introduce undefined or complex scenarios.
In the polynomial function \(g(x) = 6x^7 + \pi x^5 + \frac{2}{3}x\), the exponents are all non-negative integers: 7, 5, and 1.

• These exponents ensure the polynomial is continuous and smooth over all real numbers.
• They prevent the introduction of mathematical operations such as division by zero or taking roots of negative numbers.
• Polynomials with non-negative integer exponents are easily differentiated and integrated, which is useful in calculus.

Non-negative integer exponents form the foundation of polynomials, ensuring they are straightforward and tractable.