Problem 17 Determine whether each expressio... [FREE SOLUTION]

Chapter 13: Problem 17

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. $$12-n+n^{4}$$

Short Answer

Expert verified

The expression is a trinomial as it has three terms and meets polynomial criteria.

Step by step solution

Identify Polynomial Terms

A polynomial is an expression composed of variables, coefficients, and exponents that are whole numbers. The terms in the expression $12-n+n^4$ are 12, $-n$, and $n^4$.

Determine if Expression is a Polynomial

Check each term to ensure they are polynomial terms: 12 is a constant (polynomial term), $-n$ is a linear term (polynomial term), and $n^4$ is a power of a variable (polynomial term). Since all terms fit the criteria, $12-n+n^4$ is a polynomial.

Classify the Polynomial

Count the number of terms in the polynomial. There are three terms: 12, $-n$, and $n^4$. A polynomial with three terms 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.

Monomial

When we talk about polynomials, a monomial is the simplest type you can encounter. It represents a single term expression consisting of a coefficient (a number) and a variable raised to a non-negative integer power. For example, if you see something like $4x^3$ or $7y$, these are monomials.
A monomial is quite straightforward:

It has no addition or subtraction in its structure; just a product of numbers and variables.
The exponent of the variable must be a whole number.

Saying $5z^0$ seems a bit strange, but remember that any number to the power of zero is one, so this is effectively just the number 5, still acting as a monomial. Understanding these characteristics helps identify what qualifies as a monomial when scanning more complicated expressions.

Binomial

A binomial is a very popular term in algebra with a distinctive two-term structure. Observe any expression called a binomial, and you'll notice it involves two terms either added or subtracted. Examples include $x + 3$ or $2y^2 - 5y$. These two terms can be:

Numbers alone, which are constants.
Variables elevated to whole-number exponents.

It's important in simplifying expressions or expanding expressions like powers of a binomial, such as $(a+b)^2$. In this scenario, the formula $(a+b)^2 = a^2 + 2ab + b^2$ shows how two terms can generate more when expanded. Binomials can make complex expressions more approachable by breaking them down into sizable chunks.

Trinomial

Trinomial expressions are essentially a step further than binomials, containing three distinct terms. They are used frequently in algebraic operations and form a key part of solving quadratic equations and factorization.
Here's what makes a trinomial:

It's composed of exactly three terms, each a product of constants and variables raised to whole number powers.
The arrangement can involve addition or subtraction between the terms.

For example, take the expression $12 - n + n^4$, where it combines a constant, a linear term, and a variable raised to the power of four. Such structure helps in applying concepts like factoring trinomials into products of binomials and solving for variable values in equations. Recognizing a trinomial sets the stage for mastering quadratic functions and beyond.

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Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. $$12-n+n^{4}$$

Short Answer

Step by step solution

Identify Polynomial Terms

Determine if Expression is a Polynomial

Classify the Polynomial

Key Concepts

Monomial

Binomial

Trinomial

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