Problem 6 - Algebra 1: Concepts and Skills

Question

Identify the polynomial by degree and by the number of terms. $$ 4 w^{3}-8 w+9 $$

Step-by-Step Solution

Answer

The given polynomial has degree 3 and consists of 3 terms.

Step 1: Determine the Degree of the Polynomial

The given polynomial is \(4 w^{3}-8 w+9\). Observe the powers of the variable \(w\) for each term. The first term has a power of 3, the second term has a power of 1 (since no power implies power is 1), and the third term has no variable attached to it, therefore it can be considered to have a power of 0. The highest power is 3. Therefore the degree of the polynomial is 3.

Step 2: Count the Number of Terms

Count the number of terms, i.e., the groups of variables and constants in the polynomial. In the given polynomial \(4 w^{3}-8 w+9\), there are three terms - \(4 w^{3}\), \(-8 w\), and \(9\). So the number of terms in the polynomial is 3.

Key Concepts

Identifying Polynomials

To understand algebra, one must be adept at identifying polynomials. A polynomial is a mathematical expression consisting of variables, also known as indeterminates, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. But what does this all mean? In simpler terms, a polynomial looks like a string of terms, such as numbers and letters, that may be added, subtracted, or multiplied together. It's like a mathematical sentence where the terms are the words. Importantly, the terms should not divide by a variable, and the exponent of a variable must always be a whole number.

For example, when we observe the expression 4w^{3}-8w+9, it qualifies as a polynomial because it meets all required conditions: it has terms that are combined using addition and subtraction, and the variable w has only non-negative integer exponents. Identifying polynomials is essential before we can further dissect them to understand their properties.

Terms of a Polynomial

Each piece of a polynomial, separated by plus or minus signs, is called a term. It's akin to how each distinct bead on a string makes up a necklace. In the polynomial 4w^{3}-8w+9, the terms are 4w^{3}, -8w, and 9. Each term represents a separate component of the polynomial.

Terms consist of the product of a coefficient (a number) and variables raised to powers. The first term 4w^{3} includes the coefficient 4 and the variable w raised to the power of 3. The second term -8w has a coefficient of -8 with w to the power of 1 (implied), and the third term 9 is actually just a coefficient without a variable, which implicitly means w is raised to the power of 0. This last term is also known as the constant term. Understanding the terms of a polynomial aids in other operations like addition and subtraction of polynomials, where like terms - those with the same variables and powers - are combined.

Polynomial Powers

The 'power' in a term of a polynomial refers to the exponent applied to the variable within that term. If we focus on the term 4w^{3}, the number 3 is the power or exponent, which indicates how many times the variable w is multiplied by itself. Powers are crucial as they give us the polynomial's degree, which is the highest power of the variable in the polynomial.

The degree of a polynomial is a fundamental characteristic that influences the shape and behavior of its graph. It also plays a role in determining potential roots and the behavior of the polynomial at extreme values. For example, in 4w^{3}-8w+9, the highest power is 3 from the term 4w^{3}, making it a third-degree polynomial. The degree can give us a preview of the polynomial's complexity; generally, the higher the degree, the more turns and changes in direction its graph will have.