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Chapter 5: Problem 23

Identify each polynomial as a monomial, binomial, trinomial, or none of these. Also, give the degree. $$ -7 y^{6}+11 y^{8} $$

Short Answer

Expert verified
Binomial, degree 8

Step by step solution

01

- Identify the number of terms

A polynomial鈥檚 classification depends on the number of terms it has. Count the terms in the polynomial \(-7y^{6} + 11y^{8}\). There are two terms.
02

- Determine the type of polynomial

With two terms, the polynomial is classified as a binomial. Therefore, \(-7y^{6} + 11y^{8}\) is a binomial.
03

- Find the degree of the polynomial

The degree of a polynomial is the highest degree of any term. The degrees of the terms in \(-7y^{6} + 11y^{8}\) are 6 and 8. The highest degree is 8. Therefore, the degree of the polynomial is 8.

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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 polynomial with just one term. These could be numbers, variables, or the product of numbers and variables with non-negative integer exponents. For example, \(5x^3\) and \(-2\) are monomials. Identifying monomials is quite simple:
  • Look for a single term with no addition or subtraction operations involved.
  • The term can include coefficients (numbers) and variables (letters) raised to powers.
If you find only one such structure in your polynomial, it is a monomial.
binomial
When a polynomial consists of exactly two terms, it is classified as a binomial. Consider the example given in the exercise: \(-7y^{6} + 11y^{8}\). This polynomial has two terms: \(-7y^6\) and \(11y^8\).
  • If you see two distinct terms joined typically by a plus or minus sign, you have a binomial.
  • Binomials are especially useful in key algebraic operations like factoring and expanding expressions.
So, if your polynomial looks something like \(3x^2 + 2\) or \(5y - y^3\), you鈥檝e encountered a binomial.
trinomial
A trinomial is a polynomial with three terms. Each term in a polynomial is separated by either a plus or minus sign. For example, \(x^2 + 3x + 2\) is a trinomial.
  • A polynomial expression that consists of exactly three separate terms is a trinomial.
  • It can often be found in expressions like quadratic equations, which are fundamental in algebra.
Whenever you encounter a polynomial that looks like \(3a^2 + 2a - 4\), you are looking at a trinomial.
degree of a polynomial
The degree of a polynomial is one of its most defining features. It is the highest exponent in the polynomial. For example, in the polynomial \(-7y^{6} + 11y^{8}\), the terms have degrees 6 and 8. The highest degree here is 8.
  • The degree provides essential information about the polynomial's behavior, especially its end behavior and the number of roots.
  • To find the degree, you simply identify the term with the largest exponent.
Understanding the degree helps in solving equations and analyzing polynomial graphs. For instance, \(2x^5 - 4x^3 + x\) is a polynomial of degree 5.

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