Understanding Polynomial Functions

What Is a Polynomial?

A polynomial is an algebraic expression made from:

• Variables
• Constants
• Coefficients
• Addition or subtraction
• Whole-number exponents

For example:

3x² + 5x − 7

is a polynomial.

It contains three terms:

3x², 5x, −7

Another example is:

2x⁴ − 3x2 + x + 6

This is also a polynomial because all the powers of x are non-negative whole numbers.

What Is a Polynomial Function?

A polynomial function is a function that can be written in the form:

f(x)=a_n​xⁿ + a_n−1​xⁿ⁻1 + ⋯ +a₂​x² + a₁​x + a₀​

where:

• a₀​, a₁​, a₂​, …, a_n​ are constants called coefficients.
• n is a non-negative whole number.
• a_n ≠ 0.

For example:

f(x) = 2x³ − 5x² + 4x − 1

is a polynomial function.

Polynomial functions can be very simple or contain many terms.

Recognising Polynomial Functions

To decide whether an expression is a polynomial, look carefully at the exponents of the variable.

The exponents must be:

0, 1, 2, 3, 4, …

In other words, they must be non-negative integers.

Polynomial Examples

f(x) = 4x³ − 2x + 7 

g(x) = x⁵ + 3x² − 8 

h(x) = 6x − 4

p(x) = 9

All of these are polynomial functions.

Notice that even a constant such as 9 can be considered a polynomial.

What Is NOT a Polynomial?

Some expressions may look similar to polynomials but do not meet the rules.

Variables in the Denominator

Consider:

f(x) = \( \frac{3}{x} \) + 2

We can rewrite:

\( \frac{3}{x} = 3x^{-1} \)

The exponent is negative, so this is not a polynomial.

Variables Inside Roots

Consider:

f(x) = \( \sqrt[]{x} \) ​+ 4

Since:

\( \sqrt[]{x} = x^{ \frac{1}{2}} \)​

the exponent is a fraction.

Therefore, this is not a polynomial.

Variables as Exponents

Consider:

f(x) = 2^x

Here, the variable is in the exponent rather than the base.

This is an exponential function, not a polynomial.

Quick Rule

A polynomial cannot have:

• Negative powers of the variable
• Fractional powers of the variable
• Variables in denominators
• Variables as exponents

Terms and Coefficients

A polynomial is made up of terms separated by addition or subtraction.

Consider:

f(x) = 5x³ − 2x² + 7x − 4

The terms are:

5x³, −2x², 7x, −4

The numerical values multiplying the variables are called coefficients.

The coefficients are:

5, −2, 7

The number −4 is the constant term.

Writing Polynomials in Standard Form

Polynomials are usually written in standard form.

This means arranging the terms from the highest exponent to the lowest exponent.

For example:

4 + 2x³ − 5x + x²

can be rearranged as:

2x³ + x² − 5x + 4​

Now the powers decrease from left to right:

3, 2, 1, 0

Writing a polynomial in standard form makes it easier to identify its important features.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable after the polynomial has been simplified.

For example:

f(x) = 4x³ + 2x2 − 7x + 1

The exponents are:

3, 2, 1, 0

The highest exponent is 3.

Therefore:

Degree = 3​

More Examples of Degree

Consider:

f(x) = 7x⁵ − 3x² + 4

The highest exponent is 5.

Therefore:

Degree = 5

Consider:

g(x) = 8x² + 2x − 1

The highest exponent is 2.

Therefore:

Degree = 2

Consider:

h(x) = 6x − 9

The highest exponent is 1.

Therefore:

Degree = 1

Leading Term and Leading Coefficient

When a polynomial is written in standard form, the first term is called the leading term.

Consider:

f(x) = 6x⁴ − 3x² + 2x − 9

The leading term is:

6x⁴​

The number multiplying the variable in the leading term is called the leading coefficient.

Therefore:

Leading coefficient = 6​

The leading term is especially important because it strongly influences the overall shape and behaviour of the polynomial's graph.

Worked Example

Consider:

f(x) = 3 − 7x² + 2x⁵ + 4x

Step 1: Write in standard form

f(x) = 2x⁵ − 7x² + 4x + 3

Step 2: Identify the degree

The highest exponent is 5.

Degree = 5​

Step 3: Identify the leading term

2x⁵​

Step 4: Identify the leading coefficient

2​

Step 5: Count the terms

There are four terms.

This simple process can be used to analyse almost any polynomial.

Classifying Polynomials by Degree

Polynomials can be given special names according to their degree.

For degrees greater than five, we often simply say sixth-degree polynomial, seventh-degree polynomial, and so on.

 

 

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Notice that different degrees can produce very different graph shapes.

Classifying Polynomials by Number of Terms

Polynomials can also be classified according to the number of terms they contain.

Monomial

A polynomial with one term.

5x³

Binomial

A polynomial with two terms.

x² − 4

Trinomial

A polynomial with three terms.

x² + 5x + 6

For polynomials with four or more terms, we usually just call them polynomials.

Combining the Classifications

We can describe a polynomial using both its degree and its number of terms.

For example:

3x² + 5x − 7

has degree 2, so it is quadratic.

It has three terms, so it is a trinomial.

Therefore, we can describe it as a:

quadratic trinomial

Another example:

4x³ − 8

has degree 3 and two terms.

Therefore, it is a:

cubic binomial

Worked Example: Classifying a Polynomial

Consider:

f(x) = 7x⁴ − 3x² + 5

Degree

The highest exponent is 4.

Therefore, it is quartic.

Number of Terms

There are three terms:

7x⁴, − 3x2, 5

Therefore, it is a trinomial.

Complete Classification

quartic trinomial​

The leading term is:

7x⁴

and the leading coefficient is:

7

Polynomial Functions and Their Graphs

Every polynomial function can be represented by a graph.

For example, a linear polynomial such as:

f(x) = x + 1

produces a straight line.

A quadratic polynomial such as:

f(x) = x²

produces a parabola.

A cubic polynomial such as:

f(x) = x³

produces a different curved shape.

 

 

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One important feature of polynomial graphs is that they are smooth and continuous. They do not suddenly break, jump, or contain sharp corners.

As we study polynomial functions further, the degree and leading coefficient will help us predict the shape of these graphs.

Polynomials in Real-World Situations

Polynomial functions can be used to model many real-world relationships.

They can appear in:

• Physics
• Engineering
• Economics
• Computer graphics
• Architecture
• Population models
• Area and volume problems

For example, the area of a square with side length x+3 is:

A = (x + 3)²

Expanding gives:

A = x² + 6x + 9

The relationship between the side length and area can therefore be represented using a quadratic polynomial.

Did You Know?

The word polynomial comes from words meaning many terms.

• poly means many.
• nomial relates to terms or names.

You may recognise the same pattern in words such as monomial, binomial, and trinomial.

Key Terms

Polynomial – An algebraic expression containing terms with non-negative integer powers of variables.

Polynomial function – A function defined using a polynomial expression.

Term – A single part of an algebraic expression separated by addition or subtraction.

Coefficient – A numerical value multiplying a variable.

Constant term – A term containing no variable.

Degree – The highest exponent in a polynomial.

Leading term – The term with the highest degree when the polynomial is written in standard form.

Leading coefficient – The coefficient of the leading term.

Standard form – Writing polynomial terms from highest degree to lowest degree.

Monomial – A polynomial containing one term.

Binomial – A polynomial containing two terms.

Trinomial – A polynomial containing three terms.

Key Takeaways

• A polynomial function contains variables raised only to non-negative whole-number powers.
• Negative or fractional powers of variables are not allowed in polynomial functions.
• Variables cannot appear in denominators or as exponents.
• Polynomials are normally written in standard form, from highest power to lowest.
• The degree is the highest exponent in the polynomial.
• The leading term is the highest-degree term.
• The leading coefficient is the coefficient of the leading term.
• Polynomials can be classified by degree as constant, linear, quadratic, cubic, quartic, and so on.
• They can also be classified by number of terms as monomials, binomials, or trinomials.
• The degree and leading coefficient will later help us understand the behaviour and graphs of polynomial functions.