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Chapter 5: Problem 33

Classify each polynomial as a monomial, a binomial, a trinomial, or a polynomial with no special name. \(x^{2}-23 x+17\)

Short Answer

Expert verified
Trinomial

Step by step solution

01

Identify the number of terms

Count the number of terms in the polynomial. The given polynomial is displayed as \(x^{2} - 23x + 17\). The terms are separated by addition or subtraction operators.
02

Determine the type of polynomial based on the number of terms

A monomial has one term, a binomial has two terms, a trinomial has three terms, and a polynomial with more than three terms has no special name. In this case, the polynomial has three terms: \(x^{2}\), \(-23x\), and \(17\).
03

Conclusion

Since the polynomial has three terms, it is classified as a trinomial.

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Key Concepts

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

Monomials
In algebra, a monomial is an expression that contains only one term. This single term is a product of a constant and a variable, which can have an exponent. For instance, 5, -3x, and 7y^2 are all examples of monomials.

Monomials are composed by:
  • A constant (e.g., 7)
  • A variable (e.g., x)
  • A product of constants and variables, including exponents (e.g., -2x^3)
Key points to remember about monomials:
  • No addition or subtraction signs are present.
  • Each term must be multiplicative, possibly with an exponent.
Understanding monomials is foundational as it simplifies complex polynomials.
Binomials
A binomial is an algebraic expression that consists of exactly two terms. These terms are separated by either an addition or subtraction operator. Examples of binomials include x + 5, 3y - 4, and 2a^2 - b.

Characteristics of binomials:
  • Exactly two distinct terms.
  • The terms can include both constants and variables.
  • Separated by addition or subtraction operators.
Remember, identifying binomials is easy: just count the terms! If there are two, it鈥檚 a binomial.
Trinomials
A trinomial is a polynomial that has exactly three terms. These terms are separated by either addition or subtraction. Examples of trinomials include x^2 + 3x + 2, y^2 - 4y + 4, and a^2 - 2a + 1.

Here鈥檚 what defines a trinomial:
  • Three distinct terms.
  • Terms can have variables, constants, or both.
  • Separated by addition/subtraction operators.
Identifying a trinomial involves counting the number of distinct terms. If there are three, it鈥檚 a trinomial. The simplicity of trinomials makes them easy to identify and work with, especially in factoring and polynomial equations.

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Most popular questions from this chapter

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{5 x^{7} y}{-2 z^{4}}\right)^{3} $$

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