Problem 11 - Algebra and Trigonometry

Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial, then list its terms and state its degree. \(x-x^{2}+x^{3}-x^{4}\)

Step-by-Step Solution

Answer

The polynomial has 4 terms: \(x, -x^2, x^3, -x^4\). Its degree is 4.

Step 1: Recognize the Type of Polynomial

The given polynomial is \(x - x^2 + x^3 - x^4\). A polynomial is called a binomial if it has two terms, a trinomial if it has three terms, and a monomial if it has one term. Since this polynomial has four terms \((x, \ -x^2, \ x^3, \ -x^4)\), it doesn't fit neatly into these categories. It is simply a polynomial with four terms.

Step 2: List the Terms of the Polynomial

The terms of the polynomial \(x - x^2 + x^3 - x^4\) are identified as follows: \(x\), \(-x^2\), \(x^3\), and \(-x^4\). Listing these separately provides the specific components of the polynomial.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable \(x\) in its terms. The terms in this polynomial are \(x^1\), \(-x^2\), \(x^3\), and \(-x^4\). Therefore, the highest power here is \(4\), indicating that the polynomial is of degree 4.

Key Concepts

Monomial

A monomial is a polynomial that consists of only one term. This term can be a constant, a variable, or a product of constants and variables raised to non-negative integer powers.

• Examples of monomials include: \(5x^2\), \(-3\), and \(xyz^3\).
• A monomial containing a constant without any variables is sometimes referred to as a literal number.

For a polynomial to be classified as a monomial, it should not have a sum (or difference) of terms, meaning all parts of the expression are connected through multiplication or division.

Binomial

A binomial is a type of polynomial characterized by having exactly two terms. Each term itself can involve variables and coefficients, but together they are grouped as a sum or difference:

• An example of a binomial is \(x + 1\) or \(3x^2 - 4x\).
• The terms in a binomial can have different degrees.

Binomials often play a significant role in algebraic operations such as factoring and expanding expressions, showing up in the famous binomial theorem.

Trinomial

As the name suggests, a trinomial is a polynomial with three distinct terms. Like monomials and binomials, the terms in a trinomial are separated by a plus or minus sign.

• Examples include \(x^2 + 3x + 2\) or \(2a^2 - 5a + 10\).
• Trinomials are frequently encountered in quadratic equations.

Understanding the nature of trinomials helps in mastering methods like factoring, which is crucial for solving quadratic equations.

Degree of a Polynomial

The degree of a polynomial is a key characteristic and is defined as the highest exponent occurring in the polynomial. It provides significant insight into the nature and behavior of the polynomial.

• For example, in the polynomial \(x^3 - 4x^2 + x + 6\), the highest exponent is 3, making it a polynomial of degree 3.
• Finding the degree helps in understanding the polynomial's graph, as it indicates the maximum number of roots and turns the graph can have.

Higher degree polynomials can have complex properties, but looking at their highest degree gives a good starting point for further analysis.