Problem 9 - Beginning and Intermediate Algebra

Question

Determine whether each is a monomial, a binomial, or a trinomial. $$a^{2} b^{2}+10 a b-6$$

Step-by-Step Solution

Answer

The given expression \(a^{2}b^{2} + 10ab - 6\) consists of 3 terms, so it is a trinomial.

Step 1: Identify the number of terms in the given expression

The given expression is: \[ a^{2}b^{2} + 10ab - 6 \] The terms are separated by "+" and "-" signs, so we can identify 3 terms in the expression, which are: 1. \(a^{2}b^{2}\) 2. \(10ab\) 3. \(-6\)

Step 2: Determine if the expression is a monomial, binomial, or trinomial

Because the expression consists of 3 terms, it is a trinomial.

Key Concepts

Monomials

A monomial is an algebraic expression that consists of only one term. When we talk about terms, we refer to the objects in an expression separated by addition or subtraction signs.
For example, in a monomial like \(3x^2\), there are no additions or subtractions involving multiple terms. It's just a single term that includes variables, exponents, and coefficients.

• The general structure of a monomial is \(ax^n\), where \(a\) is the coefficient, \(x\) is the variable, and \(n\) is the exponent.
• Monomials can also be constants, like number 7, since it is one term without a variable.

Understanding monomials is a foundational skill in algebra, important not only for understanding simple expressions but also for grasping more complex polynomial expressions.

Binomials

A binomial is a polynomial that has exactly two terms. The prefix "bi-" means two, which makes it easy to remember. These terms are often chained together with a plus or minus sign.
A basic structure of a binomial looks like \(ax^m + bx^n\), where \(a\) and \(b\) are coefficients, and \(m\) and \(n\) are non-negative integers representing the exponents.

• Binomials are considered a step up in complexity from monomials because they involve operations between two terms.
• An example of a binomial is \(3x^2 + 5\).

Grasping how to handle binomials is crucial for solving equations, factoring, and performing algebraic manipulations.

Trinomials

Trinomials are polynomials made up of three terms. The prefix "tri-" nods to the number three, just like a triangle has three sides.
Consider the expression \(a^2b^2 + 10ab - 6\) as mentioned in our exercise. It contains three individual terms: \(a^2b^2\), \(10ab\), and \(-6\).

• Trinomials often have a more complex structure and can be written as \(ax^2 + bx + c\), typical for quadratic equations, where \(a\), \(b\), and \(c\) are coefficients.
• These expressions provide more room for interaction between variables and often require more involved methods for solving, such as factoring or using the quadratic formula.

Understanding trinomials is a gateway to higher-level algebra and calculus, as it often involves exploring solutions to quadratic equations and engaging in mathematical modeling.