Introduction to Polynomials

by Carson
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Sometimes, we see equations without the = sign, and we have to simplify it. How? So, I’ll give you an introduction to polynomials in this article.

The Definition of Polynomials

Polynomials are terms that are combined using the addition, subtraction, multiplication, division by integers, or positive exponentiation. Also, they can have any number of terms, including 1. (Even a constant can be a polynomial, but is a monomial.) The text below is an example.

\[\sqrt {34} + 55^2 + p\] \[x^{108} \over {22+86}\] But, even these are valid: \[3 + 4\] \[7\] However, you can neither simplify nor expand the lower two, right?

Terms

Okay, then, what are the bits and pieces in polynomials? Well, they are terms. The number is the main part of the term, but don’t forget the symbol preceding (not succeeding) it. We split them into like terms and unlike terms because analyzing the terms are essential for solving equations like that.

Like Terms and Unlike Terms

So, what are like terms? They are variables that have the same variable and power. In fact, they’re the keys to csimplifying polynomials since they can be added or subtracted, and turn into a single term. For instance:

\[7x + 33m – 8x\] \[7x – 8x + 33m\] Therefore, \[33m – x\] is the answer.

However, though, unlike terms can’t be simplified. Think of a way to simplify 3x + 4y. So, you will be confused and find out this isn’t the right way if you try to do that.

Coefficients

So, what about the front part of a term if it has a variable? Well, that’s not the symbol. They’re coefficients. For instance, the coefficient for 9y is 9. They are usually numbers or even a sole constant, but they can be variables if there aren’t any integers or decimals in the term. For example, the coefficient for ‘ay’ is ‘a’. Yet, there is something that we haven’t mentioned yet, and we will show that in the next example.

\[7x – 1.5x + 77x \over {11^3}\] If you think that the coefficient for the module after 7x is 1.5, you’re wrong.

So, remember to include the + or – sign to the coefficient.

How to Simplify or Expand Polynomials?

Well, the simplest operation of polynomials is combining like terms. Want to know more? Scroll back to the Like Terms and Unlike Terms session. Also, you can remove the brackets by modifying the arithmetic symbols. For example:

\[44 – (-7 + vy)\] \[44 + 7 + vy\]

You need to know about some simple formulas so that you can solve the equations. These are what you already knew, right?

\[x (y + z) = xy + xz\] \[x (y – z) = xy – xz\] \[{x + y \over {z}} = {x \over z} + {y \over z}\] \[{x – y \over {z}} = {x \over z} – {y \over z}\]

Let’s practice that. For example:

\[(x + 1)(2x – 3)\] \[x(2x – 3) + 1(2x – 3)\] Therefore, \[2x^2 – 3x + 2x – 3\] \[2x – x – 3\]

In that example, we applied the first formula so that we can solve that. So, try to make your questions and practice the skills.

So, in this article, we had an introduction to polynomials, we talked about terms and coefficients and discussed how to simplify or expand polynomials.

References, Credits, and Links

  1. (n.d.). Definition of Coefficient – Math is Fun. Retrieved November 15, 2020, from https://www.mathsisfun.com/definitions/coefficient.html
  2. (n.d.). Polynomials – Math is Fun. Retrieved November 15, 2020, from https://www.mathsisfun.com/algebra/polynomials.html
  3. (n.d.). Term Definition (Illustrated Mathematics Dictionary) – Math is Fun. Retrieved November 15, 2020, from https://www.mathsisfun.com/definitions/term.html

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