Table of Contents
What is a Polynomial Function?
Polynomial Function – Definition: A polynomial function is a mathematical function that can be expressed as a sum of terms, each of which is a monomial multiplied by a coefficient. In other words, a polynomial function is a function that can be written in the form:
f(x) = a 0 xn + a 1 xn-1 + a 2 xn-2 + … + a n-1 x + a n
where a 0 , a 1 , a 2 , …, a n are real numbers and n is a positive integer.
Polynomial Function Definition
A polynomial function is a mathematical function that can represented by a sum of terms, each of which is a product of a number (the coefficient) and a power of x (the variable). The degree of a polynomial is the largest power of x in any of its terms. Polynomial functions are used to model a wide variety of real-world situations.
A polynomial function can written in standard form, using the variable x, the degree of the polynomial, and the coefficients of the terms in descending order of power:
f(x) = a 0 xn + a 1 xn-1 + a 2 xn-2 + … + a n-1 x + a n
where a 0 is the coefficient of the x0 term, a 1 is the coefficient of the x1 term, and so on.
Polynomial functions can also written in factored form, using the factors of the polynomials greatest common divisor (GCD). The GCD is the largest number that divides evenly into all the terms of the polynomial. The factors of the GCD are the polynomial’s roots, or solutions. In factored form, a polynomial function can written as:
f(x) = (x − r 1 )(x − r 2 )(x − r 3 ) … (x − r n )
Polynomial Function Examples
A polynomial is a mathematical function that can written in the form:
f(x) = a n xn + a n-1 xn-1 + … + a 1 x + a 0
Where a 0 , a 1 , a 2 , …, a n are real numbers and x is a real variable.
Polynomials can used to model a wide variety of situations. Some examples include:
- The height, h(t), of a ball thrown into the air at time, t
- The temperature, T(x), of a room as a function of time, x
- The cost, C(x), of producing x units of a product
All of these situations can represented by polynomial functions.
Types of Polynomial Function
There are three types of polynomial function: linear, quadratic, and cubic.
Linear Polynomial: A linear polynomial is a polynomial with degree one. The equation of a linear polynomial is y = ax + b, where a and b are constants.
Quadratic Polynomial: A quadratic polynomial is a polynomial with degree two. The equation of a quadratic polynomial is y = ax2 + bx + c, where a, b, and c are constants.
Cubic Polynomial: A cubic polynomial is a polynomial with degree three. The equation of a cubic polynomial is y = ax3 + bx2 + cx + d, where a, b, c, and d are constants.
Graphs of Polynomial Function
The graphs of polynomial functions are continuous and smooth. They typically rise and then level off as they near infinity. The shapes of their graphs depend on the degree of the polynomial function. Higher degree polynomial functions have more complex shapes.
Quadratic Polynomial Function
A quadratic polynomial function is a function that can represented by a polynomial equation of the form
where “a”, “b”, and “c” are real numbers and “x” is a real number. The graph of a quadratic polynomial function is a parabola.
Graph of High Degree Polynomial Function
This graph of a high degree polynomial function shows the general shape of the curve. As the degree of the polynomial increases, the curve becomes more and more curved.
