# Particular Kinds of Polynomials

(a) **Monomials:** A monomial is a polynomial with only one term. For example,

\[\begin{array}{l}p\left( x \right):3x\\q\left( y \right):{\pi ^2}\end{array}\]

(b) **Binomials:** A binomial is a polynomial with two terms. For example,

\[\begin{array}{l}p\left( x \right):\pi x + 2{x^2}\\q\left( y \right):{y^{100}} - 1\\r\left( z \right):\sqrt 2 + {z^3}\end{array}\]

(c) **Trinomials:** A trinomial is a polynomial with three terms. For example,

\[\begin{array}{l}p\left( x \right):a{x^2} + bx + c\\q\left( y \right):{y^3} + {y^4} + {y^{1000}}\\r\left( z \right):\pi + \sqrt 2 z + \sqrt 3 {z^2}\end{array}\]

What about the following expression?

\[p\left( x \right):{x^2} + \sqrt 2 + \sqrt 3 \]

Is this a binomial or a trinomial? You may say that this is a trinomial, since it has three terms. However, strictly speaking, the correct answer is that this is a binomial, because *it has only two terms with respect to the exponent of* *x*. This means that if you consider the polynomial from the perspective of exponents of *x*, you will see that there are only two powers of *x* occurring in this polynomial, namely 2 and 0:

\[p\left( x \right):{x^2} + \left( {\sqrt 2 + \sqrt 3 } \right){x^0}\]

Thus, we have to count the *different* powers of *x* to decide how many terms it has. As another example, consider the polynomial

\[q\left( y \right):y + \sqrt 2 + \sqrt 3 + \sqrt 5 \]

We may be tempted to think that this is a *quadrinomial* (a polynomial with four terms), but in fact, there are only two terms in it with different powers of the variable *y*:

\[q\left( y \right):{y^1} + \left( {\sqrt 2 + \sqrt 3 + \sqrt 5 } \right){y^0}\]

So this will actually be a binomial.