Problem 15 - Algebra 2

Question

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ 3 z^{2}-5 z+11 $$

Step-by-Step Solution

Answer

Yes, it is a polynomial with a degree of 2.

Step 1: Identification of Polynomial

A polynomial is an expression that consists of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. The given expression is \( 3z^2 - 5z + 11 \). Each term follows the rules of a polynomial: \(3z^2\) with exponent 2, \(-5z\) with exponent 1, and constant \(11\) with exponent 0. Therefore, it is a polynomial.

Step 2: Determining the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the expression. In the polynomial \(3z^2 - 5z + 11\), the term with the highest degree is \(3z^2\), which has an exponent of 2. Hence, the degree of this polynomial is 2.

Key Concepts

degree of a polynomial

The degree of a polynomial tells us the highest power of the variable that appears in the expression. It's an important aspect because it indicates the polynomial's behavior, such as the number of roots or the general shape of its graph. For example, in the polynomial expression \(3z^2 - 5z + 11\), you observe several terms, and each term might have different powers of the variable \(z\).

• The term \(3z^2\) has a power of 2.
• The term \(-5z\) has a power of 1 (since \(z = z^1\)).
• The constant \(11\) has a power of 0 (since all constants can be seen as \(11z^0\)).

The degree of the polynomial is determined by the highest of these powers, which in this case is 2. This means \(3z^2\) significantly influences the behavior of the entire polynomial, especially as \(z\) becomes very large or very small.

identifying polynomials

A polynomial is a specific type of algebraic expression that has certain features. To determine if a given expression is a polynomial, you need to check the presence of variables, coefficients, and their arrangement. The key rules to follow are that the exponents of the variables must be non-negative integers, and the operations permitted are addition, subtraction, and multiplication.
In the expression \(3z^2 - 5z + 11\):

• Each term, like \(3z^2\), \(-5z\), and \(11\), adheres to these rules.
• The exponents (2, 1, and 0) are non-negative integers.
• Only addition and subtraction connect the terms.

Hence, \(3z^2 - 5z + 11\) is identified as a polynomial because it meets all these criteria.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra, allowing us to construct equations, inequalities, and functions. These expressions can involve:

• Variables, like \(z\).
• Numerical coefficients, like 3 and -5.
• Operations such as addition \(+\), subtraction \(-\), and multiplication \(\times\).
• Exponents, which are often integers in algebraic expressions.

The expression \(3z^2 - 5z + 11\) is a typical example of an algebraic expression. It contains the variable \(z\), coefficients 3 and -5, a constant term 11, and involves operations of addition and subtraction. Recognizing these elements helps in manipulating and understanding more complex algebraic problems.