Problem 16
Question
State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(6 x^{4}+3 x^{2}+4 x-8\)
Step-by-Step Solution
Verified Answer
Degree: 4, Leading Coefficient: 6.
1Step 1: Identify and Verify Polynomial
First, examine the given expression: \(6x^4 + 3x^2 + 4x - 8\). Notice that it consists of terms with variable \(x\) raised to whole number exponents, and it does not involve division by a variable or negative exponents. Therefore, this is a valid polynomial.
2Step 2: Determine the Highest Exponent
In a polynomial, the degree is the highest exponent of the variable. Here, the terms have exponents \(4\), \(2\), \(1\), and the constant term is equivalent to \(0\). The highest exponent is \(4\).
3Step 3: Identify the Leading Coefficient
The leading coefficient is the coefficient of the term with the highest degree. In the polynomial \(6x^4 + 3x^2 + 4x - 8\), the term with the highest degree is \(6x^4\). Therefore, the leading coefficient is \(6\).
Key Concepts
Degree of a PolynomialLeading CoefficientSingle-Variable Polynomial
Degree of a Polynomial
Understanding the degree of a polynomial is essential for analyzing and simplifying these mathematical expressions. The degree is determined by the highest power of the variable present in the polynomial equation. In simpler terms, it's the largest exponent found in the terms of the polynomial. For example, in the polynomial \(6x^4 + 3x^2 + 4x - 8\), the term with the highest exponent is \(6x^4\), thus the degree of this polynomial is \(4\).
When identifying the degree, it is important to consider only whole number exponents. Negative or fractional exponents, or any terms involving variables in the denominator, indicate that the expression is not a true polynomial. Remember, the degree can help you understand the polynomial's behavior and its graph, such as determining the number of roots and how the graph grows at both ends.
When identifying the degree, it is important to consider only whole number exponents. Negative or fractional exponents, or any terms involving variables in the denominator, indicate that the expression is not a true polynomial. Remember, the degree can help you understand the polynomial's behavior and its graph, such as determining the number of roots and how the graph grows at both ends.
Leading Coefficient
The leading coefficient is another vital part of understanding polynomials. It is defined as the coefficient of the term with the highest degree in a polynomial. Taking our example polynomial \(6x^4 + 3x^2 + 4x - 8\), the term with the highest degree is \(6x^4\). Here, the coefficient is \(6\), making it the leading coefficient.
Why does the leading coefficient matter? By looking at the leading coefficient, you can predict how the polynomial behaves as the variable grows larger. Specifically, the sign of the leading coefficient affects the direction of the polynomial's graph. A positive leading coefficient means the graph eventually goes upwards, while a negative one indicates it goes downwards. It's a powerful piece of information when sketching or analyzing polynomial functions, particularly concerning end behavior.
Why does the leading coefficient matter? By looking at the leading coefficient, you can predict how the polynomial behaves as the variable grows larger. Specifically, the sign of the leading coefficient affects the direction of the polynomial's graph. A positive leading coefficient means the graph eventually goes upwards, while a negative one indicates it goes downwards. It's a powerful piece of information when sketching or analyzing polynomial functions, particularly concerning end behavior.
Single-Variable Polynomial
A single-variable polynomial is one of the simplest types of polynomials, yet understanding it lays the foundation for working with more complex types. As the name suggests, it involves a polynomial expression in just one variable. In our exercise, \(6x^4 + 3x^2 + 4x - 8\) is an example of such a polynomial as it only involves the variable \(x\).
The presence of one variable simplifies the analysis since each term corresponds to a straightforward product of that variable raised to an integer power, multiplied by a coefficient. When examining whether an expression is a single-variable polynomial, ensure there are no multiple variables interacting within the same expression. This simplification makes tasks like finding the degree and leading coefficient straightforward, as demonstrated in the solved exercise.
The presence of one variable simplifies the analysis since each term corresponds to a straightforward product of that variable raised to an integer power, multiplied by a coefficient. When examining whether an expression is a single-variable polynomial, ensure there are no multiple variables interacting within the same expression. This simplification makes tasks like finding the degree and leading coefficient straightforward, as demonstrated in the solved exercise.
Other exercises in this chapter
Problem 16
Solve each equation. State the number and type of roots. \(x^{5}-8 x^{3}+16 x=0\)
View solution Problem 16
Factor completely. If the polynomial is not factorable, write prime. $$ 12 c d^{3}-8 c^{2} d^{2}+10 c^{5} d^{3} $$
View solution Problem 16
Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ x^{3}-9 $$
View solution Problem 16
Simplify. $$ \left(a^{3} b^{2}-a^{2} b+2 a\right)(-a b)^{-1} $$
View solution