Particular Kinds of Polynomials

Particular Kinds of Polynomials

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(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.

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