Problem 77 - Algebra and Trigonometry

Question

What is a polynomial function?

Step-by-Step Solution

Answer

A polynomial function is a mathematical expression consisting of variables and coefficients, executed through addition, subtraction and multiplication. Each term of the polynomial (monomial) is of the form \( ax^n \), where \(a\) is a constant, \(x\) is the variable, and \(n\) is a non-negative integer. The highest power of \(x\) is called the degree of the polynomial.

Step 1: Defining a Polynomial

A polynomial is a mathematical expression consisting of variables and coefficients that are combined using addition, subtraction and multiplication. No operation of division by a variable is considered in a polynomial. Also, any exponent of the variable must be a whole, non-negative number.

Step 2: Structure of Polynomial

It is represented in the form \(ax^n + bx^{n-1} + cx^{n-2} + ... + zx + y\) , where \(a, b, c, ... , z, y\) are constants and \(n\) is a non-negative integer. Each term of the polynomial must be of the form \( ax^n \) where \(a\) is a constant, \(x\) is the variable and \(n\) is a non-negative integer. Each term is called a monomial and the highest power of x in the polynomial is called the degree of the polynomial.

Step 3: Examples of Polynomial

Examples of polynomials are: 1. Linear Polynomial: \( ax + b \), where \(a\) and \(b\) are constants and \(a ≠ 0\).2. Quadratic Polynomial: \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants and \(a ≠ 0\).3. Cubic Polynomial: \( ax^3 + bx^2 + cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a ≠ 0\).