Problem 46 - Elementary and Intermediate Algebra

Question

Find the degree of each polynomial. See Example \(1 .\) $$ \frac{7}{8} t^{10}-\frac{1}{8} t^{16} $$

Step-by-Step Solution

Answer

The degree of the polynomial is 16.

Step 1: Identify the Individual Terms

The given polynomial is \( \frac{7}{8}t^{10} - \frac{1}{8}t^{16} \). It consists of two terms: \( \frac{7}{8}t^{10} \) and \( -\frac{1}{8}t^{16} \).

Step 2: Find the Degree of Each Term

The degree of a term in a polynomial is the exponent of the variable in that term. For \( \frac{7}{8}t^{10} \), the degree is 10; for \( -\frac{1}{8}t^{16} \), the degree is 16.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms when arranged in descending order. In the polynomial \( \frac{7}{8}t^{10} - \frac{1}{8}t^{16} \), the highest degree is 16.

Key Concepts

Understanding Polynomial Terms

A polynomial is a mathematical expression composed of multiple terms, each consisting of variables and coefficients. In the polynomial \( \frac{7}{8}t^{10} - \frac{1}{8}t^{16} \), we can identify two distinct terms: \( \frac{7}{8}t^{10} \) and \( -\frac{1}{8}t^{16} \). Each term is an individual component of the polynomial, and together they form the whole expression.

To understand a term in a polynomial, look specifically at:

• The coefficient: This is the numerical factor multiplying the variable, such as \( \frac{7}{8} \) or \( -\frac{1}{8} \) in our example.
• The variable: This represents an unknown value, commonly written as \( t \) or \( x \) for polynomials.
• The exponent: This indicates how many times the variable is multiplied by itself.

Understanding these components helps break down the polynomial into manageable parts.

Decoding Exponents in Polynomials

Exponents are a critical feature of polynomials as they define the power to which a variable is raised in each term. This is vital for determining both the degree of individual terms and the entire polynomial.

Consider the terms from our example:

• In \( \frac{7}{8}t^{10} \), the exponent is \( 10 \), which means the variable \( t \) is multiplied by itself 10 times.
• In \( -\frac{1}{8}t^{16} \), the exponent is \( 16 \), meaning \( t \) is raised to the 16th power.

The exponent is crucial because it determines the term's degree. Remember, the higher the exponent, the more significant the impact of that term within the polynomial. Thus, when comparing terms, the term with the larger exponent is typically more influential.

Arranging Terms in Descending Order

When dealing with polynomials, arranging terms in descending order of their exponents is a standard practice. This means you list the terms starting from the one with the highest exponent to the lowest.

For the polynomial \( \frac{7}{8}t^{10} - \frac{1}{8}t^{16} \), we recognize:

• The term \( -\frac{1}{8}t^{16} \) should come first, as it has a higher exponent of \( 16 \).
• Followed by \( \frac{7}{8}t^{10} \), given its lower exponent of \( 10 \).

Arranging polynomials in this way makes it easier to quickly identify the polynomial's degree. The degree of the polynomial is defined by the highest exponent present, which is \( 16 \) in our example. This systematic arrangement aids in comparing and understanding the influence of various terms.