Lesson Objectives:

At the end of this lesson you will be able to:

• Define polynomial function.
• Identify terms related to definition of polynomial function.
• Identify the given algebraic expression is polynomial or not.
• Find the sum, difference and product of two or more polynomial functions.
• Find the quotient of two polynomial functions.

Key Terms

• leading coefficient
• leading term
• Degree
• Dividend
• Divisor
• Quotient
• Remainder

Brainstorming Questions:

Classify the following functions as constant function, linear function, quadratic function or none of these:

a. $\quad f(x) = x + 4 $
b. $\quad g(x) = \frac{2}{3} – x $
c. $f(x) =
\begin{cases}
2x – 3, & \text{if } x \geq 0 \\
x + 1, & \text{if } x < 0
\end{cases}$

d. 𝑓(𝑥) = 𝑥⁹ + 2𝑥 – 5

e. ℎ(𝑥) = -𝑥² + 5𝑥 + 9

f. ℎ(𝑥) = (𝑥 + 3)(𝑥 + 1)

g. $f(x) = -\sqrt{5x} + \sqrt{2}$

h. $ h(x)=5$

i. $k(x)=5 + 4𝑥^2$

2.1 Definition of Polynomial Function

Definition

Let 𝑛 be non-negative integer and let 𝑎_𝑛, 𝑎_𝑛-1. . . 𝑎₂, 𝑎₁, 𝑎₀ be real numbers
with 𝑎_𝑛 ≠ 0. The function 𝑝(𝑥) = 𝑎_𝑛𝑥^𝑛 + 𝑎_𝑛-1𝑥^𝑛-1+ . . . + 𝑎₂𝑥² + 𝑎₁𝑥 +𝑎₀ is called a polynomial function in one variable 𝑥 of degree n.

In the above definition of polynomial function

• 𝑎_𝑛, 𝑎_𝑛-1. . . 𝑎₂, 𝑎₁, 𝑎₀ are called the coefficients of the polynomial function (or simply the polynomial).
• The number 𝑎_𝑛 is called the leading coefficient of the polynomial and 𝑎_𝑛𝑥^𝑛
  is the leading term.
• The number 𝑛 (the exponent of the highest power of 𝑥) is the degree of the
  polynomial.
• The number 𝑎₀ is called the constant term of the polynomial.

Domain of Polynomial function: The domain of a polynomial function is the set of
all real numbers

Note:

• The constant function 𝑓(𝑥) = c, c ≠ 0 is polynomial function with degree zero.
• The constant function 𝑓(𝑥) = 0 is called the zero polynomial with no degree
  assigned to it.

Example 1:

Find the degree, leading coefficient and constant term of the following polynomial
functions.

1.  𝑓(𝑥) = – 10
2. 𝑓(𝑥) = 𝑥⁹+ 𝑥⁴ – 2
3. 𝑓(𝑥) = 3𝑥³ + 9𝑥² – 7𝑥 + 3/2
4. ℎ(𝑥) = -4𝑥¹⁰⁰ + 8𝑥³ + 2𝑥² – 3𝑥 + 11
5. 𝑔(𝑥) = 3𝑥³ – 𝑥⁴ + 4𝑥⁶ + 5𝑥
6. 𝑔(𝑥) = 2(3𝑥⁴ + 5𝑥² – 2) + 2𝑥 + 𝑥⁴ + 1
7. l(𝑥) = 0.

Solution:

1. It is a constant polynomial with degree 0, leading coefficient -10 and constant term -10.
2. It is a polynomial function with degree 9, leading coefficient 1 and constant term -2.
3. It is a polynomial function with degree 3, leading coefficient 3 and constant term 3/2.
4. It is a polynomial function with degree 100, leading coefficient -4 and constant term 11.
5.  It is a polynomial function with degree 6, leading coefficient 4 and constant term 0.
6. Rearrange the polynomial function 𝑔 as 𝑔(𝑥) = 2(3𝑥⁴ + 5𝑥² – 2) + 2𝑥 + 𝑥⁴ + 1

= 6𝑥⁴ + 10𝑥² – 4 + 2𝑥 + 𝑥⁴ + 1

= 7𝑥⁴ + 10𝑥² + 2𝑥 -3

It is a polynomial function with degree 4, leading coefficient 7 and constant term -3.

g. It is a zero polynomial function with no degree, no leading coefficient and constant term 0.

Example 2:

Which of the following are polynomial functions?

1. f (x) = x⁶ – 2x⁵ + x⁴ + 7x² – 9x
2.  f (x) = x⁴⁹ + 1
3. f (x) = 6x³ + 9x^–2 + x + 4
4.  f (y) = 23 – 45y
5. f (x) = 201x¹⁰⁰⁰
6. f (x) = x² – x⁵ + p
7. f (x) = | x² +x+1|
8. $f(x) =\frac{x}{2x}$
9. f (x) = (1 +2x²)(x – 3).

Solution:

(a), (b), (d), (e), (f), (g), (i) are polynomials

Definition:

A polynomial expression is an expression of the form 𝑎_𝑛𝑥^𝑛 + 𝑎_𝑛-1𝑥^𝑛-1+ . . . + 𝑎₂𝑥² + 𝑎₁𝑥 +𝑎₀ where 𝑛 is non negative integer and 𝑎^𝑛 ≠ 0. Each individual expression 𝑎_𝑘𝑥^𝑘 making up the polynomial is called a term.

Example 3:

Consider the expression  $\frac{5x^4 – 2x^3 + 8}{5} – x^4 + x^2$

a. Is it a polynomial expression?
b. Find the degree, leading coefficient and the constant term.
c.What is the coefficient of 𝑥²?

Solution:

First, write  $\frac{5x^{4} – 2x^{3} + 8}{5} – x^{4} + x^{2}$ =$ \frac{5x^{4} – 2x^{3} + 8 – 5x^{4} + 5x^{2}}{5}$ = $\frac{-2x^{3} + 5x^{2} + 8}{5}$.                                  

1. Yes, it is a polynomial expression.
2. The degree is 3, the leading coefficient is -2/5 and the constant term is 8/3.
3. 5

2.2. Operations on Polynomial Functions

Brainstorming Questions:

1.   Which of the following pairs contain like terms?
a. 7𝑥 and 𝑥                          b. 5b² and 2𝑎²
b. 3𝑥⁵ and 2𝑥⁴                     d. 2𝑥² and 2𝑦²
e. 6 and                             f. 5𝑥¹¹ and 6𝑥¹¹

Answer for brainstorming questions: (a), (d), (f) are like terms.

2.2.1. Addition of Polynomial Functions

Definition

The sum of two polynomial functions 𝑓 and 𝑔 is written as 𝑓 + 𝑔 and is defined
as: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) for all real numbers 𝑥.

Note:

• The sum of two polynomial functions is found by adding the coefficients of like
  terms.

Example:

In each of the following, find the sum of f (x) and g (x):

Remark:

1. If 𝑓(𝑥) and 𝑔(𝑥) have different degrees, the degree of 𝑓(𝑥) + 𝑔(𝑥) is the same as
   the degree of 𝑓(𝑥) or the degree of 𝑔(𝑥) whichever has the highest degree.
2. If 𝑓(𝑥) and 𝑔(𝑥) have the same degree, the degree of the sum may be lower than
   or equal to the common degree.

3. The sum of two polynomial functions is a polynomial function

2.2.2. Subtraction of Polynomial Functions

Definition

The difference of two polynomial functions 𝑓 and 𝑔 is written as 𝑓 – 𝑔, and is defined as (𝑓 – 𝑔)(𝑥) = 𝑓(𝑥) – 𝑔(𝑥) for all real numbers x.

Example 4:

In each of the following, find 𝑓 – 𝑔:

a. 𝑓(𝑥) = 5𝑥⁵ – 5𝑥³ + 6𝑥 + 4 and 𝑔(𝑥) = -𝑥⁵ + 12𝑥³ + 8𝑥 – 7.
b. 𝑓(𝑥) = 6𝑥¹⁵+ 5𝑥⁷ + 2𝑥⁴ – 4𝑥² + 3𝑥 – 7 and 𝑔(𝑥) = 5𝑥¹³+ 𝑥⁷ + 4𝑥² – 3𝑥 – 3.

Solution:

1. 𝑓(𝑥) – 𝑔(𝑥) = (5𝑥⁵ – 5𝑥³ + 6𝑥 + 4) – (-𝑥⁵ + 12𝑥³ + 8𝑥 – 7)

= 6𝑥⁵ + 17𝑥³ – 2𝑥 + 11

b. 𝑓(𝑥) – 𝑔(𝑥) = (6𝑥¹⁵+ 5𝑥⁷ + 2𝑥⁴ – 4𝑥² + 3𝑥 – 7) – (5𝑥¹³+ 𝑥⁷ + 4𝑥² – 3𝑥 – 3)

= 6𝑥¹⁵ – 5𝑥¹³+ 4𝑥⁷ + 2𝑥⁴ – 8𝑥² + 6𝑥 – 4.

2.2.3. Multiplication of Polynomial Functions

Definition:

• The product of two polynomial functions 𝑓(𝑥) and 𝑔(𝑥) is written as 𝑓 ∙ 𝑔, and is defined as:(𝑓 ∙ 𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥) for all real numbers x.
• The product of two polynomials 𝑓(𝑥) and 𝑔(𝑥) is found by multiplying each term of one by every term of the other as shown in the following example.

Example 5:

let f (x) = 3x – 2 and g (x) = 2x + 5. Then the product of f (x) and g(x) is a polynomial function:

f(x) . g (x) = (3x – 2)(2x + 5) = 3x(2x + 5) -2(2x + 5)

= 6x² + 15x – 4x – 10

= 6x² + 11x -10

Example 6:

1. Find 𝑓(𝑥) ∙ 𝑔(𝑥) where 𝑓(𝑥) = 2𝑥 + 3 and 𝑔(𝑥) = 𝑥² – 3𝑥 + 1.

a. Find 𝑓(𝑥) ∙ 𝑔(𝑥).
b.Find the degree of 𝑓, 𝑔 and 𝑓 ∙ 𝑔.

c. Is the degree of 𝑓 ∙ 𝑔 equal to the sum of the degrees of 𝑓 and 𝑔?

Solution:

1. a.𝑓(𝑥) ∙ 𝑔(𝑥) = (2𝑥 + 3)(𝑥² – 3𝑥 + 1)

= 2𝑥(𝑥² – 3𝑥 + 1) + 3(𝑥² – 3𝑥 + 1) (Distributive property)
= ( 2𝑥)( 𝑥²) + (2𝑥)(-3𝑥) + (2𝑥)(1) + (3)( 𝑥²) + (3)(-3𝑥) + (3)(1) (Distributive property)

= 2𝑥³ – 6𝑥² + 2𝑥 + 3𝑥² – 9𝑥 + 3
= 5𝑥³ + (-6𝑥² + 3𝑥²) + (2𝑥 – 9𝑥) + 3
= 6𝑥³ – 3𝑥² – 7𝑥 + 3

b. Degree of 𝑓 is 1; degree of 𝑔 is 2 and degree of 𝑓 ∙ 𝑔 is 3.
c. Yes.

Note:

1. For any two non-zero polynomial functions 𝑓 and 𝑔, the degree of 𝑓 ∙ 𝑔 is 𝑚 + 𝑛 if the degree of 𝑓 is 𝑚 and the degree of 𝑔 is n

2. If either 𝑓 or 𝑔 is the zero polynomial then 𝑓 ∙ 𝑔 becomes the zero polynomial and has no degree.

3. The product of two polynomial functions is a polynomial function.

Example 7:

A person wants to make an open box by cutting equal squares from the corners

 of a piece of metal 160 cm by 240 cm as shown in Figure below. If the edge of

each cut-out square is x cm, find the volume of the box, when x = 1 and x = 3.

Solution:

The volume of a rectangular box is equal to the product of its length,
width and height. From the Figure, the length is 240 – 2x, the width is 160 – 2x, and the height is x. So the volume of the box is

v (x) = (240 – 2x) (160 – 2x) (x)

= (38400 – 800x + 4x²) (x)
= 38400x – 800x² + 4x³(a polynomial of degree 3)
When x = 1, the volume of the box is

v (1) = 38400 – 800 + 4 = 37604 cm³
When x = 3, the volume of the box is
v (3) = 38400 (3) – 800 (3)2 + 4 (3)3

= 115200 – 7200 + 108

= 108,108 cm³

2.2.4. Division of Polynomial Functions

A number that takes the form 𝑎/b where 𝑎 and 𝑏 are integers and 𝑏 ≠ 0 is called a
rational number. If 𝑏 is positive integer, we can divide 𝑎 by 𝑏 to find two other integers q and 𝑟 with 0 ≤ 𝑟 < 𝑏 such that 𝑎/𝑏 = 𝑞 + 𝑟/𝑏.

Here, 𝑎 is called the dividend, 𝑏 is called the divisor, 𝑞 is called the quotient and 𝑟 is called the remainder.

For example, to find 𝑞 and 𝑟 when 43 is divided by 5, you usually use a process called long division as follows:

Hence, 43 ÷ 5=  = 8 + .

Here, 43 is the dividend, 5 is the divisor, 8 is the quotient and 3 is the remainder.
In almost a similar way, we can divide one polynomial by another polynomial.

Definition:

The division (Quotient) of two polynomial functions 𝑓 and 𝑔 is written as 𝑓 ÷ 𝑔, and is defined as (𝑓 ÷ 𝑔)(𝑥) = 𝑓(𝑥) ÷ 𝑔(𝑥) for all real numbers 𝑥 and 𝑔(𝑥) ≠ 0(zero polynomial).

Example 8:

Divide 3x⁴ – 3x² + 6x + 2 by x – 1.

Example 9:

Divide 4x³ – 3x + 5 by 2x – 3.

Solution: