Problem 97 - College Algebra

Question

What is a polynomial in \(x ?\)

Step-by-Step Solution

Answer

A polynomial in \(x\) is an algebraic expression that can be expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where each \(a_i\) is a real number coefficient, \(x\) is a variable, and \(n\) is the degree of the polynomial, which is a non-negative integer.

Step 1: Defining a Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can be expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_2, a_1, a_0\) are the coefficients that can be any real numbers, \(x\) is the variable, and \(n\) is a non-negative integer which represents the degree of the polynomial.

Step 2: Understanding Terminology

Coefficients are the numbers by which the variables (denoted as 'x' in this case) are multiplied. The highest power of the variable in the polynomial is termed as the 'degree'. For example, in the polynomial \(2x^3\), 2 is the coefficient and 3 is the degree.

Step 3: Examples of Polynomials

Some examples of polynomials in a single variable \(x\) are: \(2x^2+3x+1\) (a quadratic polynomial), \(5x^4-7x^2+x+9\) (a quartic polynomial), and \(x-10\) (a linear polynomial). Note that a polynomial can have any number of terms, and its degree is the highest power of the variable that appears in the polynomial.