Question

Is the expression a polynomial? If it is, give its degree. If it is not, state why not. $$ 10 z^{2}+z $$

Short answer

The expression is a polynomial with a degree of 2.

Step-by-step solution

Identify the terms of the expression

The given expression is: \[10z^{2} + z\]Identify each term in the expression. In this case, the terms are \(10z^{2}\) and \(z\).

Determine if the expression is a polynomial

A polynomial is an expression consisting of variables and coefficients, involving only non-negative integer exponents. Check each term to ensure the exponents are non-negative integers. - The term \(10z^{2}\) has an exponent of 2, which is a non-negative integer.- The term \(z\) can be written as \(z^{1}\), which has an exponent of 1, a non-negative integer.

Determine the degree of the polynomial

If the expression is a polynomial, the degree of the polynomial is the highest exponent of the variable. From the terms \(10z^{2}\) and \(z\), the highest exponent is 2.

Conclusion

Since all the exponents are non-negative integers, the given expression is a polynomial. The highest exponent is 2, so the degree of the polynomial is 2.

Key concepts

polynomial degree

The degree of a polynomial is one of its most important features. Simply put, the degree of a polynomial is the highest power (exponent) of the variable in the polynomial. For example, in the polynomial expression \(10z^{2} + z\), we have two terms: \(10z^{2}\) and \(z\). The exponents of the variable \(z\) in these terms are 2 and 1, respectively. Therefore, the highest exponent is 2, making the degree of the polynomial 2.
Understanding the polynomial degree is vital because it tells us a lot about the behavior of the polynomial. Specifically:

• The degree determines the number of roots (solutions) the polynomial can have, which is at most equal to its degree.
• The degree affects the graph's shape, especially the number of turns or 'bends' the curve will have.
• In simple terms, the higher the degree, the steeper the polynomial can behave.

Knowing how to identify the degree helps in understanding and solving more complex polynomial equations.

non-negative integer exponents

To be classified as a polynomial, an algebraic expression must contain non-negative integer exponents. This means all exponents should be whole numbers (0, 1, 2, 3, etc.) and should not be negative or fractions.
Let's consider our example again: \(10z^{2} + z\). The term \(10z^{2}\) has an exponent of 2, and \(z\) has an implied exponent of 1. Both of these are non-negative integers, which qualifies the expression as a polynomial.
For an expression to fail this criterion and hence not be a polynomial, it might have:

• Negative exponents, such as \(z^{-2}\)
• Fractional exponents, like \(z^{1/2}\)

Remember, polynomials are simple and straightforward algebraic expressions that follow this basic rule to maintain their polynomial status.

algebraic expressions

Understanding algebraic expressions is key to grasping polynomials. An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). Polynomials are a specific type of algebraic expression that adheres to strict rules.
Our example of \(10z^{2} + z\) is an algebraic expression. It combines variables (\(z\)) and coefficients (10 and 1) using addition. However, for it to be a polynomial, only non-negative integer exponents are allowed.
Other common algebraic expressions that are not polynomials include:

• Rational expressions, like \(\frac{1}{z+1}\)
• Trigonometric expressions, involving functions like \(\text{sin}z\) or \(\text{cos}z\)
• Exponential expressions, such as \(e^{z}\)

By focusing on identifying the nature of each algebraic expression, students can easily determine whether they are dealing with polynomials or not, which is a fundamental skill in algebra.