Problem 8 - Algebra and Trigonometry

Question

In Exercises 5–8, find the degree of the polynomial. $$ x^{2}-8 x^{3}+15 x^{4}+91 $$

Step-by-Step Solution

Answer

The degree of the polynomial \(x^{2}-8x^{3}+15x^{4}+91\) is 4.

Step 1: Identify the Terms

First, identify the individual terms in the equation, which are separated by the plus or minus signs. Here, the polynomial consists of the following terms: \(x^{2}\), \(-8x^{3}\), \(15x^{4}\), and \(91\).

Step 2: Identify the Exponents

The next step is to identify the exponents in each of the terms: the exponent of \(x^{2}\) is 2, the exponent of \(-8x^{3}\) is 3, and the exponent of \(15x^{4}\) is 4. The constant term \(91\) has an implied exponent of 0.

Step 3: Determine the Highest Exponent

After identifying the exponents, determine which is the highest. In this case, the highest exponent is 4, which is the exponent of the term \(15x^{4}\).

Key Concepts

Exponents

Exponents are a critical part of understanding polynomials and other mathematical expressions. They tell us how many times a number, or in this case a variable, is multiplied by itself. For example, in the term \(x^2\), the number 2 is the exponent, indicating that \(x\) is multiplied by itself: \(x \times x\). Each term in a polynomial expression can have a different exponent.

When dealing with polynomials, identifying the exponents is crucial in determining the degree of the polynomial. Each exponent represents the power to which a variable is raised. This makes it easy to spot the leading terms and understand their contribution to the polynomial’s overall structure.

It's also important to remember that constant terms, like 91 in the expression, have an exponent of 0. This is because any number raised to the power of 0 is equal to 1, and thus doesn't change the constant itself.

Polynomial Terms

Understanding polynomial terms is essential for anyone studying algebra. A polynomial is composed of several terms, each formed by a coefficient, a variable, and an exponent. For example, in the polynomial \(x^{2} - 8x^{3} + 15x^{4} + 91\), there are distinct terms: \(x^2\), \(-8x^3\), \(15x^4\), and \(91\).

Each term is separated by addition or subtraction signs. The terms \(x^2\), \(-8x^3\), and \(15x^4\) include the variable \(x\) raised to different powers. The varying exponents show how each term differs in its influence on the polynomial's overall shape on a graph.

A polynomial’s terms also define its degree, which is determined by the term with the highest exponent. In this case, \(15x^4\) has the highest exponent of 4, making the polynomial a degree 4 polynomial.

Constant Term

The constant term in a polynomial is the term that does not contain any variable, meaning its exponent is 0. In the polynomial expression \(x^{2} - 8x^{3} + 15x^{4} + 91\), the constant term is 91. This term can have a significant impact on the overall value of the polynomial, especially when evaluating the polynomial at specific values of \(x\).

While the constant term's exponent is 0, it plays a passive role in determining the degree of the polynomial. However, it is crucial for finding y-intercepts on the graph of the polynomial, as it represents the value of the polynomial when \(x = 0\).

Understanding constant terms helps in recognizing that a polynomial degree is not reliant on these terms. Instead, it’s determined by the other terms where the variable has an exponent greater than zero.