Problem 20 - Algebra 2

Question

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ \frac{4}{3} y^{2}+\frac{5}{6} y^{7} $$

Step-by-Step Solution

Answer

The expression is a polynomial with degree 7.

Step 1: Identify the Terms

The given expression is \( \frac{4}{3} y^{2} + \frac{5}{6} y^{7} \). This expression consists of two terms: \( \frac{4}{3} y^{2} \) and \( \frac{5}{6} y^{7} \).

Step 2: Evaluate the Form of Each Term

Both terms \( \frac{4}{3} y^{2} \) and \( \frac{5}{6} y^{7} \) have real coefficients and positive integer exponents (2 and 7), which makes them polynomial terms. A polynomial term has the form \( a x^n \), where \( a \) is a real number and \( n \) is a non-negative integer.

Step 3: Confirm Polynomial Status

Since both terms in the expression fit the format of polynomial terms, we can conclude that the entire expression \( \frac{4}{3} y^{2} + \frac{5}{6} y^{7} \) is a polynomial.

Step 4: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the expression. In this expression, the highest exponent is 7 (from \( \frac{5}{6} y^{7} \)), so the degree of the polynomial is 7.

Key Concepts

Degree of a Polynomial

A polynomial's degree is an important concept. It indicates the highest power of the variable in the expression. This is straightforward to identify: simply look at the term with the largest exponent.
For example, in the expression \( \frac{4}{3} y^{2} + \frac{5}{6} y^{7} \), the exponents are 2 and 7.
The term with the exponent 7, \( \frac{5}{6} y^{7} \), determines the polynomial's degree.
Thus, the degree of this polynomial is 7.
Understanding the degree of a polynomial helps in:

• Determining the behavior of the polynomial at extreme values (positive or negative infinity).
• Analyzing the growth and end behavior of the graph of the polynomial.
• Comparing different polynomials and their potential intersections.

Polynomial Terms

Polynomial terms are individual components of a polynomial separated by addition or subtraction signs. Each term is comprised of a coefficient, a variable, and an exponent.
These terms can vary in complexity and size.
In our example \( \frac{4}{3} y^{2} + \frac{5}{6} y^{7} \), we have two terms:
- \( \frac{4}{3} y^{2} \)
- \( \frac{5}{6} y^{7} \)
Each term follows the general polynomial term structure \( a x^n \), where:

• \( a \) is the coefficient (can be any real number)
• \( x \) is the variable
• \( n \) is a non-negative integer representing the power of the variable

Polynomials can have one term (monomials), two (binomials), or more (trinomials, polynomials).
The versatility of polynomial terms allows them to model a wide variety of real-world situations.

Real Coefficients

Real coefficients in polynomials are simply the real number multipliers attached to each term's variable. They provide the weight or magnitude of that term.
Coefficients can be positive, negative, integers, or fractions, as long as they fall on the real number line.
In the polynomial \( \frac{4}{3} y^{2} + \frac{5}{6} y^{7} \), the coefficients are \( \frac{4}{3} \) and \( \frac{5}{6} \).
Some key aspects about real coefficients:

• They contribute to the term's overall size and direction (positive or negative).
• They are essential in determining the exact position and shape of a polynomial's graph.
• Unlike coefficients, imaginary numbers (numbers involving \( i \), the square root of -1) are not part of real coefficients and aren't included in standard polynomials.

Understanding real coefficients helps in building accurate mathematical models and finding solutions in various branches of science and engineering.