Problem 30 - Algebra 2

Question

Write each polynomial in standard form. Then classify it by degree and by number of terms. $$ \frac{3 x^{5}+4 x}{6} $$

Step-by-Step Solution

Answer

The polynomial in standard form is \(\frac{1}{2}x^{5} + \frac{2}{3}x\). It is a 5th-degree polynomial and it has 2 terms.

Step 1: Simplify the Polynomial

Divide each term of the polynomial \(\frac{3 x^{5}+4 x}{6}\) by the denominator to simplify it.

Step 2: Write in Standard Form

Arrange the terms in descending order of the exponents to express the polynomial in standard form.

Step 3: Classify by Degree

The degree of a polynomial is the highest exponent of the variable. Identify the degree of the polynomial.

Step 4: Classify by Number of Terms

Count the number of terms in the polynomial to classify it by the number of terms.

Key Concepts

Simplifying Polynomials

Polynomials can often appear complex when they include fractions or multiple terms, but simplifying them makes it easier to work with. To simplify a polynomial like \(\frac{3 x^{5}+4 x}{6}\), start by dividing each term by the denominator independently. This breaks the fraction into smaller parts, making it more manageable. When we split this fraction, it simplifies to \(\frac{3 x^{5}}{6} + \frac{4 x}{6}\), which then simplifies further to \(\frac{1}{2} x^{5} + \frac{2}{3} x\).
It's important to note that simplifying polynomials by eliminating fractions not only cleans up the expression but also prepares it for other operations such as addition, subtraction, and even further classification. It's a foundational skill that will recur in many areas of algebra and calculus, so taking the time to master it now pays off in the long run.

When simplifying polynomials, always look to combine like terms, factor out common elements, and reduce fractions where possible. This step is crucial for solving equations, graphing polynomial functions, and understanding the fundamental aspects of polynomial expressions.

Polynomial Degree Classification

The degree of a polynomial is a very important concept as it tells us about the behavior of the graph of the polynomial function, such as how many turns it might have. It is fundamentally the highest exponent of the variable within the polynomial. After simplifying \(\frac{3 x^{5}+4 x}{6}\) to \(\frac{1}{2} x^{5} + \frac{2}{3} x\), we see that the highest exponent of \(x\) is 5. Consequently, this polynomial is classified as a fifth-degree polynomial.

Knowing the degree of a polynomial helps in predicting the number of roots and the end behavior of its graph. For instance, a fifth-degree polynomial could potentially have five roots and will have an odd-numbered end behavior, indicating that the two ends of the graph go opposite directions. It also suggests a certain number of maximum turns or changes in direction the graph can have. Being able to classify a polynomial by its degree is essential for these further analyses and for comprehending the structure and characteristics of polynomial functions.

Polynomial Terms Classification

Classifying a polynomial by the number of its terms is crucial as it gives insight into the form and complexity of the expression. After simplifying and writing \(\frac{3 x^{5}+4 x}{6}\) in standard form as \(\frac{1}{2} x^{5} + \frac{2}{3} x\), we can see that there are two terms in this polynomial. Terms are separated by addition or subtraction symbols, and each term is a product of a number, called a coefficient, and a variable raised to a power.

In this case, the polynomial is called a binomial because it contains two terms. If a polynomial has one, two, or three terms, it is called a monomial, binomial, or trinomial, respectively. Having more terms will simply classify it as a polynomial without a special name associated with the number of terms. This classification assists in identifying the type of polynomial you are dealing with and can dictate the approach to solving or manipulating the expression in relation to algebraic operations, factoring, and other mathematical processes.