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Chapter 13: Problem 40

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. $$\frac{1}{5 a^{2}}$$

Short Answer

Expert verified
The expression \( \frac{1}{5a^2} \) is not a polynomial because it involves division by a variable.

Step by step solution

01

Identifying the Expression

The expression given is \( \frac{1}{5a^2} \). We need to see if it fits the definition of a polynomial.
02

Definition of a Polynomial

A polynomial is an expression consisting of variables (also called indeterminates), coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. No division by a variable is allowed.
03

Analyzing the Expression

The expression \( \frac{1}{5a^2} \) involves division by \( a^2 \), which is a variable. This indicates that the expression does not meet the criteria for a polynomial, since division by variables is not allowed in polynomials.
04

Conclusion

Since \( \frac{1}{5a^2} \) involves division by a variable \( a^2 \), it cannot be classified as a polynomial of any kind, whether monomial, binomial, or trinomial.

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Key Concepts

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

Monomial
A monomial is one of the simplest forms of polynomial expressions. It consists of just a single term. This term could be a constant, a variable, or the product of constants and variables. Here are some important characteristics of a monomial:

  • Only one term: Monomials have only one term, which means no addition or subtraction within the expression.
  • Non-negative integer exponents: The variables in monomials have exponents that are non-negative integers.
  • No division by variables: Monomials cannot include variables in the denominator.
Examples include expressions like 7, -3x, or 4a虏. Despite their simplicity, monomials form the building blocks for understanding more complex polynomial expressions.
Binomial
A binomial, as the prefix 'bi-' suggests, is a polynomial with two distinct terms. These terms are typically connected by either an addition or subtraction sign. Let鈥檚 delve into the specifics:

  • Two terms: Binomials have exactly two terms, leading to simple expressions like x + 7 or 5a - 2b.
  • Simpler expressions: Despite having two terms, binomials are still relatively simple and manageable compared to larger polynomials.
  • Non-negative integer exponents: Each variable in a binomial will have a non-negative integer exponent.
These can be powerful when expanded or factored and are fundamental in operations like distribution and factoring.
Trinomial
Trinomials are slightly more complex than binomials, consisting of three distinct terms. They are still classified under polynomial expressions but offer a bit more complexity:

  • Three terms: The presence of three distinct elements, such as x虏 + 5x + 6, is what defines a trinomial.
  • Interconnected through addition/subtraction: These terms are linked by addition or subtraction, providing a broader range to explore.
  • Non-negative integer exponents: Trinomials, like all polynomial expressions, follow the rule of only non-negative integer exponents.
Trinomials are particularly interesting because they can often be factored into products of binomials, making them pivotal in algebra.
Non-negative Integer Exponents
One crucial requirement for any polynomial term is that the exponents must be non-negative integers. This rule impacts how a polynomial expression is structured, and why some expressions don't qualify as polynomials. Here鈥檚 why it matters:
  • Non-negative: This means that the exponents can be zero or any positive whole number. They cannot be negative or fractions.
  • Ensures polynomial status: If any term in an expression has a negative exponent, the expression is immediately considered non-polynomial.
  • Dictates term behavior: Non-negative exponents guarantee that the polynomial doesn鈥檛 involve division by variables, aligning with the fundamental polynomial structure.
Such a requirement ensures the integrity of polynomial operations and is a critical filter when determining if an expression can be classified as a monomial, binomial, or trinomial.

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