# Introduction to Polynomials

A polynomial is an expression made up by adding and subtracting monomials [1 2]. Also the concepts of the degree and the leading coefficient of a polynomial are introduced. A polynomial in standard form as well constant, linear, quadratic, and cubic polynomials are also defined. Examples, questions and their solutions are presented.

## Monomials

We start by defining a monomial as a term of the form \[ \large \color{red}{ a x^n } \] where \( x \) is a variable, \( a \) is a constant and \( n \) is a nonnegative integer.

### Examples of Monomials

1. \( 2 x^2 \)2. \( - 3 x \)

3. \( \dfrac{1}{2} x^7 \)

## Binomials

We now define a binomial as a sum/difference of 2 monomials that are not like.

### Examples of Binomials

1. \( 2 x + 8 \)2. \( - x^3 + 3 x \)

3. \( \dfrac{1}{2} x^2 - x \)

## Trinomials

A trinomial as a sum/difference of 3 monomials that are not like.

### Examples of Trinomials

1. \( 2 x^3 + 8 x - 2 \)2. \( -\dfrac{1}{6} x^4 - 5 x - 9 \)

3. \( 0.2 x^2 - x + 4 \)

## Polynomials

A polynomial in \( x \) is the sum of any number of monomials and has the following form

\[ \large \color{red}{ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + ... + a_1 x + a_0 } \]

where the coefficients \( a_k \) are constant. If coefficient \( a_n \) is not equal to \( 0 \), then \( n \) (the highest power) is the degree of the polynomial and \( a_n \) is the leading coefficient.

### Examples of Polynomials

1. \( - \sqrt 2 x^3 + 4 x^2 - \sqrt 3 x + 12 \), leading coefficient \( - \sqrt 2 \) and degree \( 3 \).2. \( \dfrac{1}{3} x^5 - x^3 - 9 x^2 \) , leading coefficient \( \dfrac{1}{3} \) and degree \( 5 \).

## Polynomials in Standard Form

A polynomial is in standard form when it is written such that the power of the variable is in descending order.

### Example

Write each of the following polynomials in standard form and determine its degree and leading coefficient.a) \( 4 x - 9 + x^2 \) , b) \( 4 x^3 - 0.03 x^4 - x^2 \) , c) \( 4 - 3 x^5 + x^2 - \dfrac{1}{2} x^3 \) , d) \( 3 x^2 - 3 - \sqrt 3 x^3\)

### Solution

a) \( \quad x^2 + 4 x - 9 \) degree \( 2 \) and leading coefficient \( 1 \)b) \( \quad - 0.03 x^4 + 4 x^3 - x^2 \) degree \( 4 \) and leading coefficient \( -0.03 \)

c) \( \quad - 3 x^5 - \dfrac{1}{2} x^3 + x^2 + 4 \) degree \( 5 \) and leading coefficient \( - 3 \)

d) \( \quad - \sqrt 3 x^3 + 3 x^2 - 3 \) degree \( 3 \) and leading coefficient \( - \sqrt 3 \)

## Equal Polynomials

Two polynomials are equal if their corresponding coefficients are all equal.

### Example

For what values of \( a, b \) and \( c \) are the polynomials \( - x^2 + 4 x - 9 \) and \( a x^2 + b x + c \) equal?### Solution

The two given polynomials are written in standard form; hence we just compare the coefficients as follows\( \color{red}{(-1)} x^2 + \color{blue}{(4)} x + \color{green}{(- 9)} = \color{red}a x^2 + \color{blue}b x + \color{green}c \)

\( a = - 1 \) , \( b = 4 \) and \( c = -9 \)

### Example

For what values of \( a, b , c \) and \( d \) are the polynomials \( - x^3 + 4 x \) and \( a x^3 + b x + c x^2 + d \) equal?### Solution

We first need to rewrite the polynomial \( a x^3 + b x + c x^2 + d \) in standard form as follows: \( a x^3 + c x^2 + b x + d \)The given polynomials are both of degree \( 3 \) but some coefficients of the terms in the polynomial \( - x^3 + 4 x \) equal to zero. We therefore rewrite the polynomial with all its terms using zeros for coefficients of missing terms as follows:

\( - x^3 + 4 x = - x^3 + 0 x^2 + 4 x + 0 \)

We now compare the polynomials

\( - x^3 + 0 x^2 + 4 x + 0 = a x^3 + c x^2 + b x + d \)

and identify the coeffieints as follows

\( a = - 1 , b = 4 , c = 0 \) and \( d = 0 \)

## Constant Polynomials

A constant polynomial has a degree equal to \( 0 \) and is therefore of the form \( b \), where \( b \) is any contant.

### Example

The following are constant polynomial: \( - 7 \;, \; 0 \;, \; 9 \;, \; - 44\)## Linear Polynomials

A linear polynomial has a degree equal to \( 1 \) and is therefore of the form \( a x + b \)

### Example

The following are linear polynomial: \( - 2 x + 5 \;, \; x + 2 \;, \; x \;, \; \dfrac{1}{2} x - 9\)## Quadratic Polynomials

A quadratic polynomial has a degree equal to \( 2 \) and is therefore of the form \( a x^2 + b x + c \)

### Example

The following are quadratic polynomial: \( - 2 x^2 + 5x - 3 \;, \; x^2 - 6 \;, \; x^2 \;, \; \sqrt 2 x^2 + x\)## Cubic Polynomials

A cubic polynomial has a degree equal to \( 3 \) and is therefore of the form \( a x^3 + b x^2 + cx + d \)

### Example

The following are cubic polynomial: \( - x^3 + 5 x^2 - 3 x + 5 \;, \; x^3 \;, \; \dfrac{x^3}{2} \;, \; \sqrt 2 x^3 - 6 x + 6\)## Questions

### Part A

Classify the following as monomials , binomials or trinomials .

\( 2 x^2 \) , \( - x^2 - 2 x^3 \) , \( -6 \) , \( - \dfrac{1}{5} x^2 + 4 x^5 - 9 \) , \( \sqrt 5 x^2 - x - 9 \) , \( \dfrac{1}{2} x^2 - x^3 + 4\) , \( x^3 + 4\)

### Part B

Write in standard form and determine the degree and the leading coefficient of each polynomial.

a) \( - x^3 + 2x^2 - 1 \) , b) \( - x^2 - \dfrac{1}{2} x^3 \) , c) \( - 0.0001 x^2 - 9 \) , d) \( -x - 9 -\dfrac{\sqrt 2}{2} x^2 \) , e) \( - \dfrac{1}{3} x^2 - x^3 - \dfrac{1}{5} x^4 \)

### Part C

Find, if possible, the unknown coefficients so that the polynomials are equal

a) \( x^2 + 2 x - 1 = a x^2 + b x + c \)

b) \( - x^3 + 2 x^2 - 1 = a x^3 + b x^2 + c x + d \)

c) \( - x^3 + 2x - \dfrac{1}{2} x^4 = a x^4 + b x^3 + c x^2 + dx + e \)

d) \( - x^5 - 2 x^2 - x + 9 = a x^4 + c x^2 + dx + c \)

### Part D

Classify the following polynomials as constant, linear, quadratic, or cubic.

a) \( - 6 x^2 - 2 x - 1 \)

b) \( - 1 \)

c) \( - \sqrt 5 x^3 + 2 x^2 - 1 \)

d) \( - 5 + 2x \)

e) \( - x - 2 x^3 - x^2 + 9 \)

f) \( - \dfrac{x^2}{ 3} \)

g) \( - 100 \)

h) \( - x^2 + 9 + \sqrt 3 x^3 \)

i) \( - x \)

## Solutions to the Above Questions

### Part A

Monomials are: \( 2 x^2 \) , \( -6 \)Binomials are: \( - x^2 - 2 x^3 \) , \( x^3 + 4\)

Trinomials are: \( - \dfrac{1}{5} x^2 + 4 x^5 - 9 \) , \( \sqrt 5 x^2 - x - 9 \) , \( \dfrac{1}{2} x^2 - x^3 + 4\)

### Part B

a) \( - x^3 + 2x^2 - 1 \) degree = \( 3 \) and leading coefficient = \( -1 \)b) \( - \dfrac{1}{2} x^3 - x^2 \) degree = \( 3 \) and leading coefficient = \( - \dfrac{1}{2} \)

c) \( - 0.0001 x^2 - 9 \) degree = \( 2 \) and leading coefficient = \( - 0.0001 \)

d) \( - \dfrac{\sqrt 2}{2} x^2 - x - 9 \) degree = \( 2 \) and leading coefficient = \( -\dfrac{\sqrt 2}{2} \)

e) \( - \dfrac{1}{5} x^4 - x^3 - \dfrac{1}{3} x^2 \) degree = \( 4 \) and leading coefficient = \( - \dfrac{1}{5} \)

### Part C

a)Both left and right polynomials are written in standard form and have all terms included.

\( x^2 + 2 x - 1 = a x^2 + b x + c \) identify the coefficients to get: \( a = 1 \; , \; b = 2 \; , \; c = - 1 \)

b)

Both left and right polynomials are written in standard form. Rewrite with all terms of the polynomial on the left, using zero for missing terms:

\( - x^3 + 2 x^2 + 0 x - 1 = a x^3 + b x^2 + c x + d \)

Identify the coefficients to get: \( a = 1 \; , \; b = 2 \; , \; c = 0 \; , \; d = -1 \)

c)

Rewrite the polynomial on the left in standard form and including missing terms using zeros.

\( - \dfrac{1}{2} x^4 - x^3 + 0 x^2 + 2 x + 0 = a x^4 + b x^3 + c x^2 + dx + e \)

Identify the coefficients to get: \( a = - \dfrac{1}{2} \; , \; b = -1 \; , \; c = 0 \; , \; d = 2 \; , \; e = 0 \)

d)

The given polynomials do not have the same degree and therefore cannot be equal.

### Part D

a) \( - 6 x^2 - 2 x - 1 \) degree = \( 2 \) and therefore quadratic

b) \( - 1 \) degree = \( 0 \) and therefore constant

c) \( - \sqrt 5 x^3 + 2 x^2 - 1 \) degree = \( 3 \) and therefore cubic

d) \( - 5 + 2x \) degree = \( 1 \) and therefore linear

e) \( - x - 2 x^3 - x^2 + 9 \) degree = \( 3 \) and therefore cubic

f) \( - \dfrac{x^2}{ 3} \) degree = \( 2 \) and therefore quadratic

g) \( - 100 \) degree = \( 0 \) and therefore constant

h) \( - x^2 + 9 + \sqrt 3 x^3 \) degree = \( 3 \) and therefore cubic

i) \( - x \) degree = \( 1 \) and therefore linear

## More References and Links to Polynomial Functions

Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5Algebra and Trigonometry with Analytic Geometry - R.E.Larson , R.P. Hostetler , B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8

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