Problem 6 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$7 x^{3}-2 x+4$$

Step-by-Step Solution

Answer

The degree of the polynomial is 3.

Step 1: Identify Exponents

Examine each term in the polynomial: \(7x^3\), \(-2x\), and \(+4\). Identify the exponent of the variable \(x\) in each term.

Step 2: Determine the Highest Exponent

Write down the exponents for each term: 3 for \(7x^3\), 1 for \(-2x\), and 0 for \(+4\). The degree of the polynomial is the highest of these exponents.

Step 3: State the Degree

The highest exponent is 3, which is attached to the term \(7x^3\). Therefore, the degree of the polynomial is 3.

Key Concepts

Polynomial Terms

A polynomial is essentially a sum of terms, where each term includes variables raised to non-negative integer exponents and may have coefficients. In the polynomial \(7x^3 - 2x + 4\), we see three distinct terms:

• First term: \(7x^3\) which combines a coefficient, 7, with a variable \(x\) raised to the third power.
• Second term: \(-2x\) consists of a coefficient -2 and the variable \(x\) raised to the first power.
• Third term: \(+4\) is a constant term where the variable \(x\) is implied to be raised to the zero power.

Each term in a polynomial can be viewed as a building block of the expression. Understanding how these terms fit together gives us clarity on how they contribute to the entire polynomial's characteristics, such as its degree.

Exponents

Exponents are critical as they dictate the power to which a variable is raised. In our polynomial \(7x^3 - 2x + 4\), each term presents a different exponent attached to the variable \(x\).

• The term \(7x^3\) contains the highest exponent, 3, indicating that the variable \(x\) is cubed.
• In \(-2x\), the exponent is 1, which means \(x\) is simply the base variable without multiplication by itself.
• The constant term \(+4\) does not have an explicit \(x\), suggesting an exponent of 0, since any number to the power of 0 equals 1.

Exponents affect the growth and shape of the polynomial graph. Higher exponents mean the graph can potentially have unique curves and intersect the axes at different rates. Recognizing these exponents allows us to determine which terms are dominant in defining the polynomial's largest behavior.

Highest Degree Term

The degree of a polynomial is defined by the term with the highest exponent. This term is often referred to as the highest degree term. In our example, \(7x^3 - 2x + 4\), the term \(7x^3\) contains the highest exponent of 3.

This means that the polynomial is of degree 3. Identifying this term is crucial because it dictates the primary characteristics of the polynomial, including the number of roots, the potential number of turning points on a graph, and the end behavior of the polynomial function.

In broader terms, knowing the highest degree term helps in understanding complex relationships within calculus and algebra, such as integration and differentiation, where degrees of polynomials play a significant role.

• The degree tells us the number of solutions or roots the polynomial equation can have.
• It also indicates how the graph of the polynomial behaves as it extends towards infinity.

Understanding the highest degree term offers insight into the polynomial's most significant elements, making it a crucial aspect in various mathematical analyses.