Introduction to Polynomials

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