Problem 8 - College Algebra

Question

$$\text {find the degree of the polynomial.}$$ $$x^{2}-8 x^{3}+15 x^{4}+91$$

Step-by-Step Solution

Answer

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

Step 1: Identifying the Powers

Examine each term of the polynomial \(x^{2}-8 x^{3}+15 x^{4}+91\). Here the powers of x are 2, 3, 4, and 0 (For the constant 91, the implied power of x is 0 as it is equivalent to \(91x^{0}\)).

Step 2: Compare Powers

Compare all the powers of x from step 1. The powers identified were 2, 3, 4, and 0.

Step 3: Selecting the Highest Power

From the comparison in step 2, we can see that the highest power or exponent of x in the given polynomial is 4.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in a single variable, like in our exercise with variable \(x\), is typically expressed in the form:\[a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\]
where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(x\) is the variable.

Polynomials are important because they appear frequently in various branches of mathematics and have a wide range of applications. They can represent everything from simple linear relationships to complex curves in algebra and calculus.

• In polynomial functions, the number of terms is determined by the number of distinct power-plus-variable combinations.
• The degree of a polynomial, such as in the given exercise, is the highest power of the variable in the expression.

Understanding polynomial functions and their structure is crucial as it helps in analyzing and predicting like trajectories, growth rates, and more.

Exponents

Exponents are a key part of understanding polynomials because they represent how many times a variable is multiplied by itself. In the polynomial \(x^2 - 8x^3 + 15x^4 + 91\), the numbers 2, 3, and 4 are exponents.

Exponents follow specific mathematical rules that make calculations easier. When dealing with exponents:

• Adding or subtracting like powers is straightforward, for example, you can add or subtract \(x^2\) terms.
• Multiplying like bases with exponents involves adding the exponents, for instance \(x^a \times x^b = x^{a+b}\).
• Dividing involves subtracting exponents, like \(x^a / x^b = x^{a-b}\).

Understanding how to manipulate exponents is critical for simplifying, calculating, and solving polynomial expressions.

Polynomial Terms

A polynomial is made up of one or more terms. Each term consists of a coefficient and a variable raised to an exponent. For example, in the term \(15x^4\), 15 is the coefficient, \(x\) is the variable, and 4 is the exponent.

Terms can vary between polynomials. Some can have just one term (monomial), like \(7x^3\), and others can have many, like the polynomial from the exercise with 4 terms.

• Terms are separated by addition or subtraction signs.
• The degree of each term is determined by the exponent.
• The term with the highest degree defines the degree of the entire polynomial.

A clear understanding of polynomial terms helps in recognizing and working with polynomials, as well as finding their degree as shown in the exercise.