Polynomial Functions

Let's play with Polynomial Functions

Before starting exploring Polynomial Functions, Let's throw a glance at what are functions, It will help to lay a strong foundation on what you are about to learn next...

🎉 Fun fact 

Polynomial equations were studied as far back as ancient Babylon, over 4,000 years ago!

What are Functions?

A function in Mathematics is a rule that one input follows to give the output. But not like others, one input must only have one output. You can see an example below

• y = x² + 4x + 2 ✅   Because when you assign x = 3 , y = 23

• y² = x + 4x       ❌    Because when you assign x = 3 , y = ±√15

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What are Polynomial Functions?

Polynomial Functions are a type of functions that has some key differences.

They are, 

•  x has only positive integer exponents (including 0).
•  x has only real coefficients (no imaginary numbers like √-5) .
•  Function has a finite number of terms.

Every function that fulfills these conditions is a polynomial function.

A polynomial function is named as f(x), g(x), h(x), etc.

• f(x) = 2x³ + 5x² −x + 3    ✅ 

• f(x) = 3x⁻⁵ +  √-2x + 4 - ...     ❌   Because,
• x⁻⁵ has a negative exponent.
• √-2x has the coefficient of √-2 which is not an integer.
• it has an infinite number of terms.

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Components of a Polynomial Function,

As polynomial functions are easy to understand and helps to understand some of the core concepts of Mathematics, it is necessary to learn about it deeply. So lets discuss about the components of a polynomial function.

• Degree  - :  The largest exponent of x in a polynomial function
• Leading Term  - :   The term that has the largest exponent of x
• Leading Coefficient - : The coefficient of the term that has the largest value of x
• Constant - : The term that has 0 as the exponent of x in a polynomial function

eg ->      In  h(x) = 7x⁵ − 2x³ + x² + 6 , write the degree, leading term, leading                                coefficient and the constant.

• Degree  = 5
• Leading Term = 7x⁵ 
• Leading Coefficient = 7
• Constant = 6

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Fair Representation of a Polynomial Function,

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₂x² + a₁x + a₀

In this fair representation of polynomial function aₙ, aₙ₋₁, a₂, a₁, a₀ are real numbers ( n ∈ ℤ₀⁺ ).

“Try writing your own polynomial—what’s its degree and leading coefficient?”

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Why  Polynomial Functions are so important?

As it is not wrong to say that Polynomial functions are everywhere, there are important. And a good way to learn something is to find how it is used in practical sessions. They are more valuable than you ever think, check these out!

• Engineering and Architecture -: Civil engineers use polynomial functions when designing and planning to build a bridge. Likewise, there are much more...
• Computer Graphics -: Curves in animations and video games are generated using polynomial functions to create smooth graphics, offering more realistic experience.
• Business and Economics  -: Companies use polynomial functions to model profit , cost and revenue. For an example a profit function will be like this, 
• Cost function is like this => C(x) = 2000x -  750,000
• Profit function is like this =>  P(x) = R(x) - C(x)
•  Medicine and Biology -: To predict the growth of global population, spread of diseases , polynomials are used.
  
  

What have we learnt?

We discussed about Functions and then Polynomial Functions and their specifications and how they are used in real world applications. Polynomial functions are a good place for newcomers to start learning about math much more..