Problem 17 - A Graphical Approach to College Algebra

Question

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. $$\sqrt{2} x^{2}+\sqrt{3} x^{6}$$

Step-by-Step Solution

Answer

Polynomial, degree 6, binomial.

Step 1: Identify the Expression

The given expression is \( \sqrt{2} x^{2} + \sqrt{3} x^{6} \).

Step 2: Check for Polynomial

A polynomial is an algebraic expression made up of terms that are only non-negative integer powers of the variable. The expression \( \sqrt{2} x^{2} + \sqrt{3} x^{6} \) contains terms where the exponents of \( x \) are both integers (2 and 6), and the powers are non-negative. We don't have any fractional or negative exponents or division by a variable, so it is a polynomial.

Step 3: Determine the Degree

The degree of a polynomial is the highest power of the variable in the expression. In \( \sqrt{2} x^{2} + \sqrt{3} x^{6} \), the term with the highest exponent is \( \sqrt{3} x^{6} \), which means the degree is 6.

Step 4: Identify Type of Polynomial

A monomial has one term, a binomial has two terms, and a trinomial has three terms. Since \( \sqrt{2} x^{2} + \sqrt{3} x^{6} \) has two terms, it is a binomial.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a key feature to understand. It refers to the highest power (or exponent) of the variable in the expression when the polynomial is expressed in its standard form. For example, in \( \sqrt{2} x^2 + \sqrt{3} x^6 \), the highest exponent is \( 6 \), so we say that this polynomial has a degree of 6.

To find the degree, you need to identify the exponents of the variable in each term and pick the largest one:

• Look at each term in the polynomial and note the exponents (the power the variable is raised to).
• The largest exponent among all the terms is the degree.

The degree gives you an idea about the polynomial's "growth" and complexity. Polynomials with higher degrees can model more complex relationships.

Monomial

A monomial is the simplest type of polynomial. It consists of just one term. A term is any number, any variable, or the product of numbers and/or variables raised to powers.

• Monomials do not have addition or subtraction operations.
• For example, \( 3x^2 \), \( 7 \), and \( -x^5 \) are all monomials since each comprises a single, non-separated term.

Monomials are often used as building blocks for more complex polynomials. They also make it easy to perform operations like multiplication or finding the degree (which is simply the exponent of the variable in a monomial).

If you encounter just one variable term, it's most likely a monomial.

Binomial

A binomial is a polynomial with exactly two terms. These terms are separated by an addition or a subtraction sign.

• Each term can be a number, a variable, or a product of both.
• For example, \( \sqrt{2} x^2 + \sqrt{3} x^6 \) and \( x - 1 \) are binomials.

The process of recognizing a binomial involves simply counting the terms:
1. Look for two separate parts of the expression, each of which should stand alone as a monomial.2. Confirm that they are joined by either a plus or minus sign.

Binomials are very common in algebra because they appear in many expressions, equations, and factoring problems.

Trinomial

A trinomial is a polynomial made up of exactly three terms. Just like binomials, the terms in a trinomial are separated by addition or subtraction signs.

• Examples of trinomials include \( x^2 + 4x + 4 \) and \( 3x^2 - x + 2 \).
• Each term is either a standalone constant or a monomial.

To identify a trinomial:1. Count the number of terms in the expression.2. Ensure there are exactly three terms, separated by either "\(+\)" or "\(-\)" signs.

Trinomials often show up in quadratic equations and play a critical role in factoring techniques. Understanding how to manipulate and factor trinomials is crucial for solving higher-level algebra problems.