Question

consider the polynomial expression \(5 x+6-3 x^{3}-4 x^{2}\) Name the coefficients of the terms. Which is the leading coefficient?

Short answer

The coefficients of the terms in the polynomial expression are -3, -4, 5 and 6. The leading coefficient is -3.

Step-by-step solution

Identify the terms and their coefficients

In the polynomial expression \(5x + 6 - 3x^{3} - 4x^{2}\), we can break it down into four terms: \(-3x^{3}\), \(-4x^{2}\), \(5x\), and \(6\). The coefficient of each term is the numerical factor multiplying each term. Therefore, the coefficients are -3, -4, 5 and 6 respectively.

Recognizing the highest degree term

The term with the highest power of x (in this case the term \(-3x^{3}\)) is called the leading term.

Identifying the leading coefficient

The coefficient of the leading term is called the leading coefficient. So, in this case, the leading coefficient is -3.

Key concepts

Leading Coefficient

Understanding the leading coefficient of a polynomial is crucial as it can reveal significant information about the polynomial's behavior. In a polynomial, the leading coefficient is the coefficient associated with the term that has the highest degree. Take for example the polynomial expression \(5x + 6 - 3x^{3} - 4x^{2}\).

When the polynomial is written in standard form, that is, terms arranged from highest to lowest degree, we identify the leading coefficient by looking at the term with the highest exponent on the variable, which in this case is \( -3x^{3} \). Here, the leading coefficient is -3, appearing with the term of degree 3. The sign and magnitude of this coefficient affect how the graph of the polynomial will look, particularly as \(x\) becomes very large or very small.

The leading coefficient dictates the end behavior of the polynomial's graph. If the leading coefficient is positive, the graph will rise to the right and, depending on the degree, either rise or fall to the left. If the leading coefficient is negative, as in our example, the graph will descend to the right and exhibit the opposite behavior to the left.

Polynomial Terms

Polynomials are composed of one or more terms, where each term is a combination of a constant coefficient and variables raised to whole-number exponents. In our given example \(5x + 6 - 3x^{3} - 4x^{2}\), the polynomial is made up of four distinct terms.

\begin{itemize}

• \( -3x^{3} \) is a term where -3 is the coefficient and \(x^{3}\) indicates the variable raised to the third power.
• \( -4x^{2} \) is another term with -4 as the coefficient and \(x^{2}\) as the squared variable.
• \(5x\) is a linear term with 5 as the coefficient and \(x\) as the variable.
• The term 6 is a constant term as it does not contain any variables.

Each term contributes to the overall shape and features of the polynomial function. When finding solutions or analyzing graph behavior, it is essential to identify and work with these individual polynomial terms.

Degree of a Polynomial

The degree of a polynomial is a fundamental characteristic that determines many of its properties, including the shape of its graph and the number of solutions it may have. To find the degree of a polynomial, one must identify the term with the highest powered exponent and note that exponent as the degree.

In our polynomial example \(5x + 6 - 3x^{3} - 4x^{2}\), the term with the highest exponent is \( -3x^{3} \). Therefore, the degree of this polynomial is 3. This is an important piece of information because it implies certain features of the polynomial function, such as the aforementioned end behavior, the maximum number of x-intercepts it might have, and the maximum number of turns the graph can take.

A polynomial's degree can also help determine the number of roots or solutions the polynomial is expected to have, according to the Fundamental Theorem of Algebra, which, in this case, indicates there can be up to three real or complex roots for the given polynomial.