This page is here to describe the reason behind why we study this math – multiplying out brackets and factoring into brackets – in grade 10.
Definition: A polynomial is an algebraic expression that has many (poly) terms (nomial). The exponent of any variable must not be negative.
Click on the image to go to the mathisfun page for examples:
Any of our whole numbers an be expressed as a polynomial. For example the number 3457 could be written with four terms, as follows:
What are polynomials good for?
Polynomials are a building block for the math required in most STEM careers: engineering, science, computer science, data science etc.
Polynomial functions are a family of functions that follow the same algebraic structure and produce a smooth curve as seen below. What is relevant for the math learner is that these functions are really quite simple. We can study them using mostly whole numbers. Functions are like machines that have inputs and outputs. Both the inputs and outputs of polynomial functions can be any real number but we can stick to just whole numbers to understand the function well. While these functions are numerically simple, they have a full range of features that are found in other functions. They have high points (local maxima) and low points (local minima), they go upwards (increasing) and downwards (decreasing), they can be more steep and less steep, they have x intercepts and one y intercept. We can use two polynomials to make a new polynomial by adding or by multipltiplying or applying one first and then the second.
There are several families of functions. It is quite straightfoward to find applications for other families such as sinusoidal functions and exponential functions. We get to study those with applications in the grade 12 precalculus and grade 12 foundations of math courses. Through learning about the numerically simple polynomials, it is easier to learn about the more numerically complex sinusoidal and exponential functions.
A neat twist in the story of mathematics occured in the 1700s. While polynomials and other functions such as sinusoidal or exponential functions were already well known, a mathematician called Brook Taylor figured out how to express sinusoidal and exponential functions as polynomials. This is really awesome. Read more at mathisfun.(Newsflash, recently discovered work tells us that the Indian Mathematician Madhava of Sangaragrama also knew this math two centuries earlier).
What do polynomials look like?
Here are some examples of graphs of polynomials. Below is the function for each expressed in two equivalent forms.
In the applet, you see the smooth curve, the polynomial written in standard form (terms added together) and the polynomial written in factored form (terms multiplied together).
In grade 10 we learn how to expand brackets (multiply out the product of terms) and how to factor a trinomial (turn the sum of terms into a product of terms).
We also analyse a polynomial order 1 (linear function) in depth, and use concepts such as 
intercept and 
intercept which transfer to all other polynomials.
In grade 11 we graph polynomials order 1 and order 2.
By grade 12, we are ready to study the whole family of polynomials.