What are Polynomials?

In math, there are some words that are practically foundations to everything else. Just like any topic, math has its own vocabulary and one of the first words you should get comfortable with in algebra is the word “polynomial”. Polynomials are expressions involving terms which may have exponents that are multiplied, added, or subtracted from each other. In the following lesson, we will look at different examples of polynomials and learn about some special types of polynomials.

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Examples of polynomials

As mentioned above, in a polynomial, terms consist of variables or constants (numbers) that are either multiplied, added, or subtracted from each other. Additionally, any exponents must be positive whole numbers. For example, each of the expressions below would qualify as a polynomial:

  • \(5x^2\)
  • \(2x^{4}-5x+11\)
  • \(\dfrac{2}{3}z^5+2xyz-xy\)

In each case above, notice that the exponents were positive integers (positive whole numbers). If an exponent is negative, then this implies you are dividing by that term (based on the definition of a negative exponent), and polynomials cannot have any division involving the variables. A polynomial can have fractions involving just the numbers in front of the variables (the coefficients), but not involving the variables.

Examples of expressions which are not polynomials

Keeping the explanation above in mind, the following are not polynomials:

  • \(x^{-2}+x-1\)
    There is a negative exponent.
  • \(xy^{\frac{1}{2}}+2\)
    The exponent is a fraction.
  • \(\log(x) + 3x^4\)
    This involves another function: \(\log(x)\).

Special types of polynomials

There are a few special types of polynomials which get their own names. Remember, we are thinking of a term as something like \(3xy\) or 5. In other words, it is one of the pieces of the polynomial that is being added or subtracted from the other pieces/terms. When counting terms, you are assuming that none of the terms are like terms.

Monomials

Monomials are polynomials with a single term. Some examples are:

  • \(-3x^4\)
  • \(10xy\)
  • \(2x\)

Binomials

Binomials have two terms. One of the most common types of binomials you will see in algebra are those like \(3x + 5\) or \(x + 4\). These involve one variable term and one constant term and are often the factors found when working with quadratic equations.

Some other examples of binomials are:

  • \(\dfrac{1}{2}x^5 + 2x\)
  • \(4x – 9\)
  • \(12x^2y – 4xy^2\)

When it comes to multiplying this type of polynomial, there is a special rule: FOIL.

Trinomials

Trinomials have three terms. One place you will see these a lot is the classic “quadratic equation” like \(2x^2 + 5x + 1 = 0\). The expression on the left-hand side of this equation is a trinomial.

Some other example of trinomials are:

  • \(-1 + 4x + 3y\)
  • \(2x^2y + 3x + 7\)
  • \(x^2 + 3x + 1\)
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Additional reading

As you continue studying algebra, you may find the following articles and guides useful:

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