Problem 10 Determine whether each is a mono... [FREE SOLUTION]

Chapter 6: Problem 10

Determine whether each is a monomial, a binomial, or a trinomial. $$16 r^{2}+9 r$$

Short Answer

Expert verified

The given expression, $16r^2 + 9r$, is a binomial.

Step by step solution

Know the definitions of monomials, binomials, and trinomials.

A monomial is a polynomial with only one term, a binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms. The terms in a polynomial are separated by addition (+) or subtraction (-) signs.

Identify the terms in the given expression.

We are given the expression: $16r^2 + 9r$. Let's identify the terms in the expression. We have two terms here, separated by an addition sign: 1. The first term is $16r^2$. 2. The second term is $9r$.

Determine if it's a monomial, binomial, or trinomial based on the number of terms.

Since there are two terms in the expression, it is a binomial.

Write the final answer.

The given expression, $16r^2 + 9r$, is a binomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomials

Monomials are the simplest type of polynomial expressions you can encounter. They consist of only one term. This single term can have numbers (also known as coefficients), variables, or exponents, all combined by multiplication but never separated by addition or subtraction. When identifying monomials, look for these features:

There is only one term, like $3x^2$ or $-7abc^3$.
No addition (+) or subtraction (-) signs break the term apart.
It could be just a number (e.g., 5), a variable (e.g., $x$), or a combination (e.g., $4y$).

Monomials are often used as building blocks for more complex polynomials. They represent single calculated values or expressions based on the example they provide.

Binomials

Binomials are expressions made up of exactly two terms. These terms are separated by either an addition or subtraction sign. In algebra, binomials are quite common and often form the basis of quadratic expressions. Here are some features to identify binomials:

There are exactly two terms, like $x + 1$ or $3a - 5b$.
Terms are split by a single addition or subtraction sign.
Each term can itself be a combination of numbers and variables.

When encountering the expression $16r^2 + 9r$, it's straightforward to see it's a binomial due to its two distinct terms separated by an addition sign. Binomials are crucial in understanding part of polynomials because they can simplify into a single expression through simplification processes or factoring.

Trinomials

Trinomials are one step above binomials, consisting of three distinct terms. These terms, like binomials and monomials, can include numbers, variables, and exponents. Understanding trinomials is pivotal since they are often seen in quadratic equations and other important algebraic expressions. Characteristics of trinomials include:

Three separate terms make up the expression, such as $x^2 + 5x + 6$.
Terms are split by either addition or subtraction signs.
Each term has its own value or expression.

Trinomials usually appear when dealing with factorization in algebra, where the goal might be to express a quadratic or cubic polynomial as a product involving trinomials. Ascending from monomials and binomials, trinomials aid in grasping the complexity of polynomial expressions.

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91影视

Determine whether each is a monomial, a binomial, or a trinomial. $$16 r^{2}+9 r$$

Short Answer

Step by step solution

Know the definitions of monomials, binomials, and trinomials.

Identify the terms in the given expression.

Determine if it's a monomial, binomial, or trinomial based on the number of terms.

Write the final answer.

Key Concepts

Monomials

Binomials

Trinomials

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