Problem 11 - Precalculus Mathematics for Calculus

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, degree 4, it’s neither a monomial, binomial, nor trinomial.

Step 1: Identify the Number of Terms

Count the number of terms in the polynomial. The polynomial is \(x - x^2 + x^3 - x^4\). There are four terms: \(x\), \(-x^2\), \(x^3\), and \(-x^4\). Thus, the polynomial contains four terms and is neither a monomial, binomial, nor trinomial.

Step 2: List the Terms

List all the terms identified in the polynomial. The terms are \(x\), \(-x^2\), \(x^3\), and \(-x^4\).

Step 3: Determine the Degree of the Polynomial

Check the degree of each term, which is the highest power of the variable in each term. The degrees of the terms are: \(x = x^1\), \(-x^2 = x^2\), \(x^3 = x^3\), \(-x^4 = x^4\). The degree of the polynomial is the highest degree among its terms, which is 4.

Key Concepts

Monomials

Monomials are the simplest kind of polynomials. A monomial is essentially a single term. This single term can be made up of numbers, variables, or the product of numbers and variables. The key characteristic of a monomial is that it does not include any addition or subtraction between different parts of it. Think of something like \(6x^2\) or \(-3y\). These are monomials because they only contain one expression of multiplication.

Monomials help form more complex polynomial expressions. The degree of a monomial is simply the sum of the exponents of the variables within it. For example, in the monomial \(5x^3y^2\), the degree is 3+2 = 5. In simple terms, you just add up all the powers.

Remember that a constant number is also a monomial. It can be seen as having a degree of zero because it does not have a variable factor. For instance, "7" is a monomial with a degree of 0 due to its lack of variables.

Binomials

A binomial is a type of polynomial that contains exactly two distinct terms. These terms are connected by either a plus or minus sign. A common example of a binomial is \(x + 1\) or \(2y - 3z\). These expressions have precisely two terms that involve addition or subtraction.

Binomials are especially useful in algebra because they can often be factored or expanded through processes like the distributive property or FOIL (First, Outer, Inner, Last) method. For instance, when multiplying two binomials together, you can use FOIL to efficiently expand the expression. This multiplication often leads to another polynomial form.

The degree of a binomial is the highest degree of its individual terms. If the terms in a binomial are \(4x^3\) and \(-7x^2\), the degree of this binomial is the highest power, which is 3.

Trinomial

Trinomials are a bit more complex than monomials or binomials, as they contain three distinct terms. For instance, the expression \(x^2 + 2x + 1\) is a trinomial. It is made up of three separate parts that are connected by addition or subtraction.

Trinomials are common in quadratic functions and equations. They often factor into simpler expressions, which is a key step in solving many algebraic equations. For example, the trinomial \(x^2 + 5x + 6\) can be factored into \((x + 2)(x + 3)\). Thus, solving problems often involves breaking down trinomials like these into simpler binomials.

Just like monomials and binomials, the degree of a trinomial is determined by looking at the highest degree present among its terms. With \(x^2 + 2x + 1\), the highest degree is 2, given by \(x^2\). This is important to remember, as it influences how you approach equations involving these expressions.