Problem 97 - College Algebra Essentials

Question

What is a polynomial in \(x ?\)

Step-by-Step Solution

Answer

A polynomial in \(x\) is a mathematical expression that can be presented in the form \(p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(n\) is a non-negative integer, and \(a_0, a_1, ..., a_n\) are the coefficients which are real numbers.

Step 1: Definition

A polynomial in \(x\) is a mathematical expression that can be written in the following general form: \(p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\). Here, \(n\) is a non-negative integer, and \(a_0, a_1, ..., a_n\) are the coefficients which are real numbers. It can also be expressed as a sum of monomials.

Step 2: Breakdown of Components

Looking at the parts of the polynomial, \(x\) is known as the variable or indeterminate, while the \(a\) values are coefficients. The highest power of \(x\) for which the corresponding coefficient is not zero is referred to as the degree of the polynomial. Each term \(a_ix^i\) of the polynomial is called a monomial.

Step 3: Examples

For example, \(p(x) = 5x^4 - 3x^2 + 2x - 7\) is a polynomial in \(x\) of degree 4 with coefficients 5, -3, 2, and -7 for \(x^4, x^2, x, and the constant term, respectively.