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Chapter 10: Problem 4

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$-12 p^{4}$$

Short Answer

Expert verified
The polynomial -12p^4 is a monomial.

Step by step solution

01

Identify the Polynomial

Identify the given polynomial: The polynomial is -12p^4
02

Count the Terms

Count the number of terms in the polynomial. Here, there is only one term -12p^4 Thus, it is a single term polynomial.
03

Classify the Polynomial

Classify the polynomial based on the number of terms: -If a polynomial has one term, it is called a monomial. -If a polynomial has two terms, it is called a binomial. -If a polynomial has three terms, it is called a trinomial. -If a polynomial has more than three terms, it is simply called a polynomial. Since the given polynomial has only one term, the polynomial -12p^4 is a monomial.

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Key Concepts

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

monomial
A polynomial with just one term is known as a monomial. The prefix 'mono' means one, making it easy to remember. Monomials can include numbers, variables, or the product of both. For example, \(5x\), \(-3\), and \(2a^2b^3\) are all monomials. They are simple to work with because there are no additional terms to confuse things. In the exercise provided, \(-12p^4\) is identified as a monomial because it consists of one single term. No addition or subtraction separates different components. Whether it is a single variable or a variable with a coefficient, as long as there is only one term, it is a monomial.
binomial
A polynomial with two terms is called a binomial. The prefix 'bi' means two, similar to a bicycle having two wheels. Binomials are a bit more complex than monomials but still fairly simple to understand. Consider examples like \(x + 1\) or \(3a - 2b\). Each of these examples has two terms. To identify a binomial:
  • Look for exactly two terms separated by a plus or minus sign
  • Each term can be a constant, a variable, or a product of both
Binomials are significant in algebra because many algebraic processes, like factoring, often involve them. In the context of the exercise, if we had a polynomial like \( -12p^4 + 3q \), it would be classified as a binomial.
trinomial
A trinomial is a polynomial with three terms. The prefix 'tri' means three, which may remind you of a tricycle with three wheels. Examples of trinomials include \(a^2 + 2a + 1\) and \(3x - 2y + 4\). Trinomials are commonly used in algebra, particularly in quadratic equations. When identifying a trinomial, look for:
  • Exactly three terms
  • Terms separated by plus or minus signs
Like binomials, each term in a trinomial can be a constant, a variable, or a product of both. Trinomials are essential for factoring and solving quadratic equations. If our original exercise had an example like \( -12p^4 + 3q - 5r \), it would be a trinomial.
terms in polynomials
Understanding the terms in a polynomial is crucial for classifying it correctly. A 'term' can be a number, a variable, or a product of both. Terms in a polynomial are separated by plus or minus signs. Here are the steps for identifying and counting terms:
  • Look for plus (+) or minus (-) signs
  • Each section separated by these signs is a term
For instance, \( 3x^2 + 2x - 1 \) has three terms: \( 3x^2 \), \( 2x \), and \( -1 \). The number of terms determines what kind of polynomial it is:
  • One term: Monomial
  • Two terms: Binomial
  • Three terms: Trinomial
  • More than three: Sometimes called 'polynomials' or 'multinomials'
Properly counting and identifying terms ensures accurate classification, such as in our initial example with \( -12p^4 \), which has just one term, making it a monomial.

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Most popular questions from this chapter

Multiply the binomials using various methods. $$(2 y-9)(5 y-7)$$

Simplify each expression. $$(n-8)\left(n^{2}-4 n+11\right)$$

Divide the monomials. $$72 p^{12} \div 8 p^{3}$$

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

Simplify each expression. $$\left(5 a^{2}+2 a-12\right)+\left(9 a^{2}+8 a-4\right)$$
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