Problem 7 Find the degree of each of the f... [FREE SOLUTION]

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Chapter 5: Problem 7

Find the degree of each of the following polynomials and deter. mine whether it is a monomial, binomial, trinomial, or none of these. See Examples I through 3. $$ 3 z x-5 x^{2} $$

Short Answer

Expert verified

The polynomial $3zx - 5x^2$ is a binomial with a degree of 2.

Step by step solution

Identify the Terms

The given polynomial expression is $3zx - 5x^2$. It consists of two terms: $3zx$ and $-5x^2$.

Find the Degree of Each Term

For the term $3zx$, the degree is found by adding the exponents of all variables. In this case, the exponents are $z^1$ and $x^1$, so the degree is $1 + 1 = 2$. For the term $-5x^2$, the variable $x$ has an exponent of 2 which is the degree of this term.

Determine the Degree of the Polynomial

The degree of a polynomial is the highest degree among all its terms. Here, the highest degree is 2, corresponding to the terms $3zx$ and $-5x^2$. So, the polynomial has a degree of 2.

Classify the Polynomial

A binomial is a polynomial with exactly two terms. Since $3zx - 5x^2$ has two terms, it is a binomial.

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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 type of algebraic expression that consists of only one term. This means a monomial does not contain any addition or subtraction signs within it. The term can be made up of numbers and variables, which are multiplied together. For example, the expression $4xy^2$ is a monomial. It is important to note that the degree of a monomial is determined by the sum of the exponents of its variables.

Key characteristics of a monomial include:

Only one single term.
No addition or subtraction within the term.
Multiplication of numbers and variables is allowed.
Degree calculated by summing up the exponents of all included variables.

For instance, in the monomial $7a^3b^2$, the degree is calculated by adding the exponents of $a^3$ and $b^2$, giving us a degree of $3 + 2 = 5$. This simple structure makes monomials easy to handle and understand.

Binomial

A binomial, like the one in the exercise $3zx - 5x^2$, is a polynomial expression that consists of exactly two terms. These terms are connected by either a plus or a minus sign. In the given example, the binomial is formed by the terms $3zx$ and $-5x^2$. Each term may contain constants, variables, or products of variables.

When dealing with a binomial:

Identify the two individual terms involved.
Calculate the degree of each term by summing the exponents of the variables within it.
The degree of the binomial is determined by the highest degree among its terms.

For $3zx$, the degree calculation involves adding the exponent of $z^1$ and $x^1$, resulting in a degree of $2$. The second term, $-5x^2$, has a degree of 2 due to the exponent on $x$. Therefore, the overall degree of this binomial is also 2.

Trinomial

A trinomial is an algebraic expression that consists of three terms. Much like monomials and binomials, trinomials are composed of numbers, variables, or their products, and each term is separated by a plus or minus sign. Trinomials are slightly more complex than binomials due to the extra term they contain.

Trinomials can be understood as follows:

Contain exactly three distinct terms.
The terms are separated by + or - signs.
The degree of each term is assessed individually by summing the exponents of its variables.
The degree of the trinomial itself is the highest degree amongst its three terms.

For example, consider the trinomial $x^2 + 2xy + y^2$. The degrees of the individual terms $x^2$, $2xy$, and $y^2$ are $2$, $1+1=2$, and $2$, respectively. Thus, the trinomial has an overall degree of 2, as the highest degree among its terms.

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91影视

Find the degree of each of the following polynomials and deter. mine whether it is a monomial, binomial, trinomial, or none of these. See Examples I through 3. $$ 3 z x-5 x^{2} $$

Short Answer

Step by step solution

Identify the Terms

Find the Degree of Each Term

Determine the Degree of the Polynomial

Classify the Polynomial

Key Concepts

Monomial

Binomial

Trinomial

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