Problem 21 - A Graphical Approach to College Algebra

Question

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. $$-5 \sqrt{z}+2 \sqrt{z^{3}}-5 \sqrt{z^{5}}$$

Step-by-Step Solution

Answer

The expression is not a polynomial because it contains fractional exponents.

Step 1: Identify Polynomial Terms

To check if the expression is a polynomial, each term should have non-negative integer exponents. Examine each term: \(-5 \sqrt{z},\ 2 \sqrt{z^3},\ -5 \sqrt{z^5}\). We see that each term includes square roots, which means the exponents are fractional.

Step 2: Convert Radicals to Exponents

Rewrite each term with fractional exponents. \(-5 \sqrt{z} = -5z^{1/2},\ 2 \sqrt{z^3} = 2z^{3/2},\ -5 \sqrt{z^5} = -5z^{5/2}\). The exponents are \(1/2, 3/2,\) and \(5/2\), which are fractions.

Step 3: Determine Polynomial Characteristics

A polynomial requires all exponents to be non-negative integers. Since \(1/2, 3/2,\) and \(5/2\) are not integers, the expression \(-5 \sqrt{z} + 2 \sqrt{z^3} - 5 \sqrt{z^5}\) is not a polynomial.

Key Concepts

Degree of a Polynomial

Polynomials are mathematical expressions involving a sum of powers of variables. The degree of a polynomial is simply the highest power of the variable in the polynomial. This degree helps us understand the shape and complexity of the polynomial graph.

For example, consider the polynomial \(3x^4 + 5x^2 + 2x + 7\). Here, the highest power of \(x\) is 4, so the degree is 4.

It's important to note that each term must have non-negative integer exponents. This means fractions or negative numbers in the exponent render an expression non-polynomial. In the given exercise with radical terms, converting those to fractional exponents, such as \(z^{1/2}\), indicates they are not polynomials since they don't have integer exponents.

Recognizing the degree of a polynomial helps classify it and anticipate characteristics like its roots and graph behavior.

Monomial

A monomial is the simplest form of a polynomial with just one term. It can be a constant, a variable, or a combination of both. Key characteristics of a monomial include:

• It contains only non-negative integer exponents.
• It is a single term, such as \(5x^3\) or simply \(7\).

Monomials can be multiplied together to form more complex polynomials. However, in the context of the original exercise, no term qualified as a monomial because all terms had fractional exponents, disqualifying them as monomials. Understanding monomials is crucial as they form building blocks for more extensive polynomial expressions.

Binomial

A binomial is a polynomial with exactly two terms. These terms can be constants, variables, or a combination of both, and they follow the rule of non-negative integer exponents. Examples of binomials include expressions like \(x + 1\) or \(2x^2 - 3x\).

Each term in a binomial is connected by either a plus or minus sign. In identifying a binomial, ensure both terms have non-negative integer powers.

In the exercise solution, the expression didn't qualify as a binomial either due to fractional exponents. For two terms to form a proper binomial, like \(x^2 + 2x\), fractions in the power are not permissible.

Trinomial

A trinomial is simply a polynomial containing three terms. Like monomials and binomials, each term in a trinomial must have non-negative integer exponents. Examples include \(3x^2 + 2x + 1\) or \(4x^3 - x + 6\).

Trinomials are essential in algebra because they appear frequently in factorization and quadratic equations. Recognizing a trinomial involves checking its structure: there should be three terms with suitable powers.

In the provided exercise, the expression did not classify as a trinomial due to its use of fractional exponents, despite having three terms. This illustrates how strictly polynomials are defined by integer-only exponents.