3.1 What is a polynomial function?
Along with the development of Algebra, which was a word taken from a medieval book written in 820 AD by a Persian mathematicians, Polynomials were also created. Polynomial comes from two Greek words: Poly and Nomos which means many and parts respectively. Polynomials are expressions that are comprised of a coefficient, a constant, variables and exponents. These terms can be combined through the four different operations namely addition, subtraction, multiplication and division with the exemption of division wherein the denominator is a variable.
This chapter will focus all about Polynomial function. This chapter is consisted of five parts, in the first part we will review about polynomial functions, like its classification: monomial, binomial, and trinomial and the terms of the function. We learned from our past lessons that polynomials are comprised of terms and the number of terms would tell us what kind of polynomial it is.
Terms are consisted of the following: variable, coefficient and exponent. Each term had their own degrees and this is based on their exponents. The highest exponent in the function is the degree of the entire function. The coefficient of the term with the highest exponent is termed as the leading coefficient.
In the second part of the chapter we will learn about Polynomial Long Division. This process is very similar to how we divide numbers. We just need to focus on the leading terms and leading coefficient of both the dividend and the divisor to start dividing.
In the proceeding part of this chapter, we will look at Polynomial Synthetic Division. Synthetic division is generally used to solve for the zeroes of a polynomial. There will be times when we will be able to get nonreal zero in synthetic division and we will also learn the significance of even degree and odd degree in division of polynomials.
We will also look at the two theorems for polynomial division: The Remainder Theorem and the Factor Theorem. These are two related theorem, in fact the Factor Theorem is based off from the Remainder Theorem.
The Remainder Theorem states that when we divide a polynomial function f(x) to xc, the remainder R will be equal to f(c). Say we’re asked to divide $4x^3  7x + 10$, by x3, even without doing the long division we can get the remainder by solving for f(3). For the Factor Theorem, it states that if we calculate f(c) and it is equal to zero, then xc is a factor of the polynomial.
What is a polynomial function?
Lessons

a)
$f\left( x \right) = {x^4} + 7{x^3}  8{x^2} + 5$

b)
$f\left( x \right) = 34{x^2}  25{x^3} + 2x  39$

c)
$P\left( x \right) = 7$


a)
$f\left( x \right) =  5{x^3} + {x^{\frac{1}{2}}}  4$

b)
$f\left( x \right) = 2{x^2}  7{x^{  1}}  3$

c)
$f\left( x \right) = {x^4} + 9029{x^3}  \sqrt {17} {x^2} + 3897$

d)
$f\left( x \right) = \sqrt {5{x^3}}  3{x^2} + 2x  4$

e)
$f\left( x \right) = 5{x^3} = \sqrt {3{x^2}} + 2x  4$

f)
$f\left( x \right) = \sqrt 3 {x^3}  \sqrt {  3} x$


a)
Finish the table below.
Polynomial Function
Degree
Type
$P\left( x \right) = c$
$P\left( x \right) = ax + b$, $a \ne 0$
$P\left( x \right) = 4a + bx + c$, $a \ne 0$
$P\left( x \right) = a{x^3} + b{x^2} + cx + d$, $a \ne 0$
$P\left( x \right) = a{x^4} + b{x^3} + c{x^2} + {d}x + {e}$, $a \ne 0$

b)
What are the names of polynomials of degrees of five through ten?
