Problem 48 - Elementary and Intermediate Algebra

Question

Find the degree of each polynomial. See Example \(1 .\) $$ 0.4 h+0.6 h^{4} c+0.6 h^{5} $$

Step-by-Step Solution

Answer

The degree of the polynomial is 5.

Step 1: Understand the Definition of Polynomial Degree

The degree of a polynomial is the highest exponent appearing in the polynomial when it is expressed in its standard form, i.e., the exponents are arranged in descending order. Here, our polynomial is: \(0.4h + 0.6h^4c + 0.6h^5\).

Step 2: Identify Each Term in the Polynomial

There are three terms in the polynomial: \(0.4h\), \(0.6h^4c\), and \(0.6h^5\). Each term has its own degree based on the exponent of the variable \(h\). Note that if there is more than one variable in a term, the degree of the term is the sum of the exponents of all variables.

Step 3: Determine the Degree of Each Term

- The first term \(0.4h\) has a degree of 1 because the exponent of \(h\) is 1. - The second term \(0.6h^4c\) has two variables: \(h^4\) and \(c\). The degree of this term regarding \(h\) is 4.- The third term \(0.6h^5\) has a degree of 5 since the exponent of \(h\) is 5.

Step 4: Identify the Highest Degree Among the Terms

Compare the degrees of each term: 1, 4, and 5. The highest degree is 5, which comes from the term \(0.6h^5\).

Step 5: State the Degree of the Polynomial

Since the highest degree among all terms is 5, the degree of the polynomial \(0.4h + 0.6h^4c + 0.6h^5\) is 5.

Key Concepts

Exponents in Algebra

In algebra, exponents are like shorthand for expressing repeated multiplication of a variable. When you see something like \(h^5\), it means "\(h\) multiplied by itself 5 times." Each exponent in a polynomial gives us crucial information. It tells us not just about how many times to multiply the variable, but it also helps define the polynomial's degree. In a polynomial, each term includes a variable raised to a power, and this power is the exponent.

Exponents make the otherwise long-winded expressions simpler and more manageable to work with. Different rules apply when manipulating exponents, like the product rule, which states \( a^m \times a^n = a^{m+n} \), and the power rule, \( (a^m)^n = a^{m \cdot n} \). Understanding these rules can make solving polynomial problems more intuitive and less time-consuming.

Polynomial Terms

Polynomials consist of several terms, which are individual expressions that are added or subtracted from each other. A term in a polynomial can be as simple as a constant, or it might involve a variable raised to an exponent. In the polynomial \(0.4h + 0.6h^4c + 0.6h^5\), each part separated by a '+' or '-' sign is a term:

• \(0.4h\)
• \(0.6h^4c\)
• \(0.6h^5\)

Each term is composed of a coefficient (the numerical part), followed by variables with exponents. For instance, in \(0.6h^4c\), '0.6' is the coefficient, while \(h^4\) and 'c' are variables. The degree of each term, in case of multiple variables, is the sum of all exponents of the variables in that term. Recognizing each term and understanding its degree help us find the degree of the entire polynomial.

Standard Form of Polynomials

A polynomial is in its standard form when the terms are written in descending order of their degrees. This means arranging the terms from the highest degree to the lowest degree. In the example polynomial \(0.4h + 0.6h^4c + 0.6h^5\), the standard form would place the term with the highest degree, \(0.6h^5\), first, because it has the highest exponent:

• \(0.6h^5\)
• \(0.6h^4c\)
• \(0.4h\)

Standard form makes it easier to identify the degree of the polynomial, which is simply the highest exponent in the polynomial. For our polynomial, when sorted correctly, the highest exponent is 5, making the degree of the polynomial 5. Writing polynomials in standard form is a foundational practice that facilitates more advanced topics in algebra and calculus.