Problem 53 - Elementary Algebra

Question

Classify \(10 x^{3}-7 x\) as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each item.

Step-by-Step Solution

Answer

Question: Identify the type of the algebraic expression \(10x^3 - 7x\), find its degree, and list the numerical coefficients of its terms. Answer: The given expression is a binomial with a degree of 3. The numerical coefficients of its terms are 10 and -7.

Step 1: Identifying Monomial, Binomial, and Trinomial

A monomial is an algebraic expression with only one term, a binomial has two terms, and a trinomial has three terms.

Step 2: Classifying the Given Expression

The given expression is \(10x^3 - 7x\). It has two terms - \(10x^3\) and \(-7x\). Since it has two terms, it is a binomial.

Step 3: Finding the Degree

The degree of a term is the highest power of the variable in the term. The degree of an algebraic expression is the highest degree among its terms. In our given expression, the degrees of the terms are 3 (from \(10x^3\)) and 1 (from \(-7x\)). The highest degree among them is 3, so the degree of the expression is 3.

Step 4: Finding the Numerical Coefficient of Each Term

The numerical coefficient of a term is the number attached to the variable in the term. In our given expression, the numerical coefficients of the terms \(10x^3\) and \(-7x\) are 10 and -7, respectively.

Key Concepts

Degree of a Polynomial

When we talk about the degree of a polynomial, we're referring to the highest power of the variable present in the polynomial. In simpler terms, it's the largest exponent you see attached to a variable in the expression.
For example, consider the polynomial expression \(10x^3 - 7x\). This expression has two terms: \(10x^3\) and \(-7x\).
The degrees of these terms are:

• \(10x^3\) has a degree of 3, because the exponent of \(x\) is 3.
• \(-7x\) has a degree of 1, since \(x\) stands for \(x^1\).

Since the highest degree among these terms is 3, we say that the degree of the polynomial \(10x^3 - 7x\) is 3. Identifying the degree is crucial because it helps in understanding the behavior of the polynomial as well as in comparing it with other polynomial expressions.

Numerical Coefficient

The numerical coefficient is simply the number that multiplies a variable in a term of a polynomial. It doesn't consider the variable itself, only the number in front of it. In math terms, if you look at a term like \(ax^n\), \(a\) is the numerical coefficient.
Let's see how this works in our given polynomial \(10x^3 - 7x\):

• The term \(10x^3\) has a numerical coefficient of 10. This means the variable \(x\) is multiplied by 10.
• The term \(-7x\) has a numerical coefficient of -7. Here, the variable \(x\) is multiplied by -7.

Recognizing numerical coefficients is important for performing operations with polynomials like addition, subtraction or factoring. It essentially tells you how much of the variable you have, like how you would count units of something in real life.

Types of Polynomials

Polynomials can be classified based on the number of terms they contain. Knowing the type of polynomial you are dealing with can help in deciding the approach for solving it or simplifying it.
Here are the basic types:

• Monomial: A polynomial with just one term. An example could be \(5x^2\).
• Binomial: A polynomial with two distinct terms. Our expression \(10x^3 - 7x\) falls under this category because it consists of two terms.
• Trinomial: A polynomial containing three separate terms. For instance, \(x^3 + 2x^2 - 5\) is a trinomial.

Understanding these basic types makes it easier to approach algebraic operations and decide on the methods for simplifying or solving the polynomials. It also sets a foundation for understanding more complex mathematical expressions.