Problem 7 - Algebra Form and Function

Question

Which of the expressions in Exercises \(7-12\) are polynomials in \(x ?\) If an expression is not a polynomial in \(x,\) what rules it out? $$ \frac{2 x^{3}}{5}-2 x^{7} $$

Step-by-Step Solution

Answer

Answer: Yes, the expression is a polynomial in x.

Step 1: Rewrite the expression

Rewrite the given expression: $$ \frac{2 x^{3}}{5}-2 x^{7} $$ We can simplify this expression by taking the fraction into account: $$ \frac{2}{5}x^3 - 2x^7 $$

Step 2: Identify the terms and check their degrees

Now, let's identify the terms of the expression and check the degrees of x in each term: \(\frac{2}{5}x^3\) is one term with degree 3. \(-2x^7\) is another term with degree 7.

Step 3: Determine if the expression is a polynomial in x

As both terms of the expression satisfy the definition of a polynomial in x, we can conclude that the expression is a polynomial in x. Both terms have non-negative integer exponents for x and the coefficients are constants. The expression: $$ \frac{2}{5}x^3 - 2x^7 $$ is a polynomial in x.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is an important concept when analyzing polynomial expressions. It represents the highest power of the variable in the expression.
For example, in the expression \(\frac{2}{5}x^3 - 2x^7\), the degrees of the terms are 3 and 7.
To determine the degree of the entire polynomial, we look at the term with the highest degree.
In this case, \(-2x^7\) has the highest exponent of 7, which makes 7 the degree of the polynomial.
Here’s a summary to keep in mind for any polynomial:

• The degree is the largest exponent of the variable in the expression.
• The degree of a polynomial can help determine its behavior and the number of roots it might have.

Terms of a Polynomial

Terms in a polynomial are building blocks that make up the polynomial expression. Each term consists of a coefficient and a variable raised to a power.
In our expression \(\frac{2}{5}x^3 - 2x^7\), there are two distinct terms: \(\frac{2}{5}x^3\) and \(-2x^7\).
In detail, the construction of a term includes:

• A coefficient, which could be a whole number, fraction, or decimal. For example, \(\frac{2}{5}\) in the first term and \(-2\) in the second term are coefficients.
• A variable part, here it is \(x\), which may be raised to a power (exponent).

Understanding terms helps in simplifying and rearranging polynomials to make calculations easier.

Exponents in Polynomials

Exponents in polynomials are the powers to which the variable is raised in each term.
In the expression \(\frac{2}{5}x^3 - 2x^7\), the exponents are 3 and 7, respectively.
Why are exponents important?

• They play a crucial role in determining the degree of the polynomial, as the highest exponent indicates the degree.
• Exponents help define the shape and properties of the graph of the polynomial function.
• Understanding exponents is key to performing arithmetic operations on polynomial expressions, such as multiplication and division.

In polynomials, exponents must always be whole numbers and should not result in negative or fractional exponents, which would disqualify the expression as a polynomial.