# 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: $- 2 x + 5 \;, \; x + 2 \;, \; x \;, \; \dfrac{1}{2} x - 9$

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