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Chapter 10: Problem 7

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$23y^{2}$$

Short Answer

Expert verified
Monomial

Step by step solution

01

Understand the terminology

A monomial is a polynomial with just one term. A binomial contains exactly two terms, a trinomial has exactly three terms, and other polynomials have more than three terms.
02

Identify the number of terms

Observe the given polynomial: \(23y^{2}\). Count the terms in the polynomial. Here, there is only one term, which is \(23y^{2}\).
03

Classify the polynomial

Since the polynomial \(23y^{2}\) has exactly one term, it is classified as a monomial.

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Key Concepts

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

monomial
A monomial is a type of polynomial that consists of a single term. This term can be a constant, a variable, or the product of a constant and one or more variables. For example, the polynomial given in the exercise, \(23y^2\), is a monomial because it only has one term.
Monomials can look like:
  • 3
  • x
  • 5xy^2
When identifying a monomial, ensure that there are no additions or subtractions separating different parts of the expression.
binomial
A binomial is a polynomial that contains exactly two terms. These terms are usually connected by addition or subtraction. For example, in the term \(a + b\) or \(2x - 5\), each expression has exactly two distinct parts, or terms.
This means you can visually identify a binomial by recognizing it has two sections being added or subtracted. Here are a few examples of binomials:
  • \(x + y\)
  • \(3a - 4b\)
  • \(7x^2 + 9\)
When you separate the terms in a binomial, make sure you never combine more than two terms.
trinomial
A trinomial is a polynomial that is made up of exactly three terms. These terms are also connected through addition or subtraction. For example, the expression \(x + 2y + 3\) is a trinomial because it has three distinct parts.
Identifying a trinomial involves counting the number of terms present and ensuring they add up to three. Here are several examples of trinomials:
  • \(x^2 + 2x + 1\)
  • \(a + b + c\)
  • \(4x^2 - 3x + 2\)
Always note that there should be clear separations between the three terms in a trinomial.
terms in polynomial
Terms in a polynomial are the different parts separated by addition or subtraction. Each term can be a number (constant), a variable, or the product of numbers and variables. For example, in the polynomial \(5x^2 + 3x - 4\), there are three terms: \(5x^2\), \(3x\), and \(-4\).
To identify the number of terms in a polynomial, simply count the distinct sections divided by plus or minus signs.
Here are some key points to remember:
  • Monomials have 1 term.
  • Binomials have 2 terms.
  • Trinomials have 3 terms.
  • Polynomials with more than 3 terms don't have a special name.

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Most popular questions from this chapter

Factor the greatest common factor from each polynomial. $$2 q^{5}-16 q^{3}+30 q^{2}$$

Factor the greatest common factor from each polynomial. $$-9 a^{5}-9 a^{3}$$

Simplify each expression. $$\frac{x^{4}}{x^{-5}}$$

Divide the monomials. $$\frac{\left(5 m^{9} n^{3}\right)\left(8 m^{3} n^{2}\right)}{\left(10 m n^{4}\right)\left(m^{2} n^{5}\right)}$$

Factor the greatest common factor from each polynomial. $$16 u-24$$
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