Problem 67 - Algebra 1

Question

Identify the leading coefficient, and classify the polynomial by degree and by number of terms. $$x-x^{3}$$

Step-by-Step Solution

Answer

The leading coefficient is -1, the degree of the polynomial is 3, and the number of terms is 2.

Step 1: Reordering Polynomials

Reorder the given polynomial in standard form: The polynomial should be written with descending powers of the variable, therefore, it becomes \( -x^{3} + x \)

Step 2: Identifying and Classifying

Identify the leading coefficient, degree, and number of terms. The leading coefficient is the coefficient of the highest degree term, which is -1 in this case. The degree of the polynomial is 3, as it is the highest exponent. The number of terms is 2, as the polynomial can be broken down into two distinct parts, separated by the symbol +.

Key Concepts

Leading Coefficient

In a polynomial, the leading coefficient is the numerical factor that is positioned in front of the term with the largest degree or the highest power of the variable. It's very important because it can tell us about the behavior of a polynomial, such as its growth as the variable increases. In the polynomial \(-x^3 + x\), after rewriting it in standard form, the leading term is \(-x^3\). The coefficient here is \(-1\).

• For the polynomial \(3x^4 + 2x^3 + x + 7\), the leading term is \(3x^4\), making the leading coefficient \(3\).
• Changing the leading coefficient can change the direction and width of a graph's main curve.

Always remember: the highest degree terms are the ones to determine the leading coefficient.

Degree of a Polynomial

The degree of a polynomial is a measure of the variable's highest exponent in a polynomial expression. It's critical because it indicates the polynomial's rate of growth and its number of potential roots. When a polynomial is written in standard form (terms ordered from highest to lowest power), its degree is determined easily.For \(-x^3 + x\), we see the term \(-x^3\) is the leading term and has an exponent of \(3\). Thus, the degree of this polynomial is \(3\). Here's a quick look at what degree tells us:

• A cubic polynomial like this one is identified by the degree \(3\) and often can have up to three real roots.
• The degree sets the maximum number of turning points for the graph. A degree \(n\) polynomial can have up to \(n-1\) turning points.

Number of Terms

The number of terms in a polynomial indicates how many distinct monomial parts it contains. These are separated by addition or subtraction symbols.Taking the polynomial \(-x^3 + x\), we identify two terms: \(-x^3\) and \(x\). Therefore, this polynomial has two terms, making it a binomial.

• "Monomial" refers to a polynomial with just one term, like \(5x^2\).
• "Binomial" indicates two terms, such as \(3x + 2\).
• "Trinomial" is used when there are three terms in a polynomial, for example, \(x^2 + 3x + 2\).

Breaking down polynomials by the number of terms helps simplify the process of factoring or expanding them.