Problem 3
Question
In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x$$
Step-by-Step Solution
Verified Answer
The function \(g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x\) is a polynomial function. Its degree is 5.
1Step 1: Identification
The function \(g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x\) is in the form of \( ax^{n} + bx^{m} + cx^{p}\), where 'a', 'b' and 'c' are coefficients and 'n', 'm', 'p' are powers of 'x'. This form aligns with the definition of a polynomial function.
2Step 2: Determination of the Degree
The degree of a polynomial function is the highest power of the variable 'x'. In the function \(g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x\), the highest power is 5 that comes from the term \(7x^{5}\). Therefore, the degree of the polynomial is 5.
Key Concepts
Degree of PolynomialCoefficientsHighest PowerIdentifying Polynomials
Degree of Polynomial
In mathematics, the degree of a polynomial function plays a crucial role in understanding its behavior and characteristics. The degree of a polynomial is simply the highest power of the variable within its expression. For instance, in the polynomial function \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), the highest power of \(x\) is 5, as seen in the term \(7x^5\).
This means that the degree of this particular polynomial is 5.
Understanding the degree helps in predicting the graph's shape and the number of its potential solutions or roots. A higher degree often implies more complexity in its behavior, including a greater number of turns in its graph.
This means that the degree of this particular polynomial is 5.
Understanding the degree helps in predicting the graph's shape and the number of its potential solutions or roots. A higher degree often implies more complexity in its behavior, including a greater number of turns in its graph.
Coefficients
Coefficients are numerical or constant factors that multiply the terms in a polynomial.
They play a significant role in determining the specific characteristics of the polynomial function. In our example, \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), each term has a coefficient:
They play a significant role in determining the specific characteristics of the polynomial function. In our example, \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), each term has a coefficient:
- The coefficient of the first term \(7x^5\) is 7.
- The coefficient of the second term \(-\pi x^3\) is \(-\pi\).
- The coefficient of the third term \(\frac{1}{5}x\) is \(\frac{1}{5}\).
Highest Power
The highest power, or the leading exponent of a polynomial, is the exponent on the variable with the largest value in the polynomial expression. This element is significant because it not only determines the degree but also heavily influences the polynomial's behavior for very large or small values of \(x\).
In \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), the highest power of \(x\) is 5. The term associated with this power is called the leading term, which is \(7x^5\) in this polynomial.
The highest power affects how the graph of the polynomial ascends or descends towards the ends, known as end behavior.
In \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), the highest power of \(x\) is 5. The term associated with this power is called the leading term, which is \(7x^5\) in this polynomial.
The highest power affects how the graph of the polynomial ascends or descends towards the ends, known as end behavior.
Identifying Polynomials
Identifying whether a function is a polynomial involves checking its structure.
A polynomial function must be a sum of terms, each consisting of a constant coefficient multiplied by a variable raised to a non-negative integer exponent.
This means no fractional or negative powers or variables in the denominator.
Considering \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), this function qualifies as a polynomial because it matches the form \(ax^n + bx^m + cx^p\):
A polynomial function must be a sum of terms, each consisting of a constant coefficient multiplied by a variable raised to a non-negative integer exponent.
This means no fractional or negative powers or variables in the denominator.
Considering \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\), this function qualifies as a polynomial because it matches the form \(ax^n + bx^m + cx^p\):
- The exponents on all the terms' variables are non-negative integers: 5, 3, and 1.
- Each term has a real number coefficient: 7, \(-\pi\), and \(\frac{1}{5}\).
Other exercises in this chapter
Problem 2
Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=x^{3}+3 x^{2}-6 x-8 $$
View solution Problem 2
Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(x^{2}+3 x-10\right) \div(x-2)$$
View solution Problem 3
Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(a\) is directly proportional to the square of \(b\)
View solution Problem 3
In Exercises \(1-8,\) find the domain of each rational function. $$g(x)=\frac{3 x^{2}}{(x-5)(x+4)}$$
View solution