Question

consider the polynomial expression \(5 x+6-3 x^{3}-4 x^{2}\) What is the degree of the polynomial?

Short answer

The degree of the polynomial \(5 x+6-3 x^{3}-4 x^{2}\) is 3

Step-by-step solution

Identify the terms of the polynomial

The polynomial \(5 x+6-3 x^{3}-4 x^{2}\) is made up of four terms: \(5 x\), \(6\), \(-3 x^{3}\), and \(-4 x^{2}\)

Find the degrees of the terms

The degree of each term is the value of the exponent. Therefore, the degrees of the terms are: 1 for \(5 x\), 0 for \(6\), 3 for \(-3 x^{3}\), and 2 for \(-4 x^{2}\)

Identify the highest degree

The degree of the polynomial is determined by the highest degree from all its terms. Here, 3 is the highest degree.

Key concepts

Polynomial Expressions

A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial might look complex at first glance, but understanding its components can simplify its appearance considerably. Polynomials encompass terms separated by plus or minus signs. These terms consist of numbers, called coefficients, and variables often represented by letters like x or y.

For instance, consider the polynomial expression \(5x + 6 - 3x^3 - 4x^2\). This expression is made up of four specific terms: \(5x\) which is a linear term, \(6\) a constant term, \( -3x^3\) a cubic term, and \( -4x^2\) which is a quadratic term. Each of these plays a different role in defining the shape and properties of the polynomial's graph. Moreover, the expression does not have all its terms in descending powers which is a common practice for ease of understanding.

Exponents in Polynomials

Exponents play a critical role in understanding polynomials. In a polynomial, each term with a variable also has an associated exponent, which indicates to what power the variable is raised. The term's degree is the value of this exponent. Exponents must be whole numbers in a polynomial, and different terms can have different exponents.

In the expression \(5x+6-3x^3-4x^2\), the associated exponents for each term are 1, 0, 3, and 2, respectively. These exponents denote the power to which the variable x is raised. The exponent of 1 in the term \(5x\) is often not written, implying that x is to the first power. For the constant term \(6\), the exponent of the variable is considered as 0, even though the term doesn't explicitly show a variable. This is because any number raised to the power of 0 is 1, making the term essential a standalone coefficient.

Terms of a Polynomial

Breaking down a polynomial into its individual parts involves identifying its terms. The terms of a polynomial are the distinct elements that make up the expression and are typically separated by plus or minus signs. Each term is a product of a coefficient and a variable raised to an exponent.

In the provided polynomial, \(5x+6-3x^3-4x^2\), we identify four terms: \(5x\), \(6\), \( -3x^3\), and \( -4x^2\). Coefficients can be positive or negative, influencing the term's contribution to the polynomial's overall value. Identifying the terms of a polynomial is a fundamental step in performing operations like addition, subtraction, and finding the degree of the polynomial, which, in this case, is determined to be 3 as the highest exponent among the terms.