Polynomial Functions

A Polynomial function is a function in which the independent variable is a non-negative integer exponent.

An algebraic term with the highest exponent is the leading term that determines the degree of the function; its coefficient is the leading coefficient.

Polynomial Functions: Factored and Expanded Forms

  • Know the Definitions
  • Look at the Structure: In factored form, you will see parentheses and multiplication. – In expanded form, you will see addition and subtraction.
  • Check for Coefficients: In factored form, check for coefficients outside brackets. Make sure these are combined properly. In expanded form, make sure the coefficients are added correctly with the variable terms.
  • Find the Degree: Both forms should have the same degree. The degree helps you see if the two forms are the same.
  • Convert Between Forms: To change from factored form to expanded form, distribute and combine like terms. To change from expanded form to factored form, look for common factors or use grouping methods.
  • Check for Roots: You can find the roots easily from the factors in the factored form. For expanded form, set the polynomial equal to zero and solve for the roots to ensure the two forms are equivalent.
  • Practice with Examples: Work through examples of both forms to build your understanding. Start with a polynomial in expanded form, convert it to factored form, and then back to see if they match.

These steps will help you tell the difference between factored and expanded forms of polynomial functions.

polynomial function factored form

Polynomial Functions: Characteristics and Graphs

A degree of a polynomial function = maximum number of x-intercepts.

A turning point is where a function changes from increasing to decreasing or vice versa.

The maximum number of turning points (local maximums and local minimums) = one less than the degree of a function.

The sign of the leading coefficient determines the end behaviour of a polynomial function:

  • Odd number degree + positive leading coefficient: Q3 – Q1
  • Odd number degree + negative leading coefficient: Q2 – Q4
  • Even number degree + positive leading coefficient: Q2 – Q1
  • Even number degree + negative leading coefficient: Q3 – Q4
end behaviour odd number degree
end behaviour even number degree

In the expanded form, the constant value = y-intercept of the function.

In the factored form, the factors help determine the x-intercepts (roots).

The order of x-intercepts (roots) is how many times that particular x-intercept is repeated.

If the x-intercept is of an even number order (2,4,6,…) – there is a bounce and a turning point at that x-intercept.

If the x-intercept is of an odd number order (1,3,5,…) – the graph curves through the x-intercept and there is no turning point at that x-intercept.

Polynomial Functions: Symmetry

Odd Function – odd symmetry – when a function is symmetrical about the origin (rotational symmetry).

Even Function – even symmetry – when a function is symmetrical about the y-axis (mirror symmetry).

Neither Even nor Odd Function – neither symmetrical about the origin nor about the y-axis.

even and odd symmetry of polynomial function

It is also possible to determine the symmetry algebraically, by testing the equation of the function.

Given the equation of the function f(x), we can test f(-x) and see if every term changes the sign to the opposite or not.

 

For example, the function  f(x)=3x^3-x  is ODD, since

f(-x)=3(-x)^3-(-x) 

f(-x)=-3x^3+x

Every sign within the function equation has changed to the opposite.

 

The function  f(x)=2x^4-3x^2+1  is EVEN, since

f(-x)=2(-x)^4-3(-x)^2+1

f(-x)=2x^4-3x^2+1

Every sign within the function equation has remained unchanged.

 

The function  f(x)=3x^3-2x^2-15  is NEITHER, since

f(-x)=3(-x)^3-2(-x)^2-15

f(-x)=-3x^3-2x^2-15

The signs of some terms have changed and others – remained the same.

NOTE: It is important to differentiate between odd/even degree and odd/even symmetry – those are different concepts, even though similar terminology is used to describe them.

Polynomial Functions: Real World Application

Polynomial functions are essential in various real-world applications, serving as powerful tools for modeling relationships across multiple fields such as engineering, physics, and finance.

In engineering and physics, they describe phenomena like motion and forces, while in computer graphics, polynomial equations define Bezier curves crucial for creating smooth transitions in digital animations.

Additionally, in economics, they model demand and supply functions, and polynomial regression helps predict stock prices, assisting investors in decision-making.

Beyond engineering and finance, polynomial functions are utilized in data analysis and environmental science. They aid in regression analysis to fit curves to data points, facilitating insights in various sectors, including healthcare. In biology, polynomial equations model population growth, and in environmental science, they represent resource management and pollutant spread.

Their versatility in applications significantly enhances advancements in technology and everyday life, highlighting the vital role of mathematics in innovation.

Check your understanding with Polynomial Functions Quiz and Even and Odd Symmetry Quiz