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How do you determine if $\dfrac{{3x}}{{5{\text{y}}}}$ is a polynomial and if so, is it a monomial, binomial, or trinomial?

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Last updated date: 24th Feb 2024
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IVSAT 2024
Answer
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Hint: The coefficient can be any real number, including $0$. The exponent of the variable must be a whole number $:{\text{ }}0,{\text{ }}1,{\text{ }}2,{\text{ }}3$, and so on. A monomial cannot have a variable in the denominator, under a radical, have a fractional exponent, or have a negative exponent.

Complete step by step answer:
The essential structure of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the result of a number and variables with an exponent.
If ${\text{p}}\left( x \right)$is a polynomial in $x$, the highest power of $x$ in ${\text{p}}\left( x \right)$is called the degree of the
polynomial ${\text{p}}\left( x \right)$.
Types of Polynomials
A polynomial of degree $1$ is called a linear polynomial.
A polynomial of degree $2$ is called a quadratic polynomial.
A polynomial of degree $3$ is called a cubic polynomial.
To find out why it's not classified, let's rewrite $\dfrac{{3x}}{{5{\text{y}}}}$:
$\dfrac{{3{\text{x}}}}{{5{\text{y}}}} = 3x \times \dfrac{1}{{5{\text{y}}}} = 3x \times 5{y^{ - 1}}$
As an expression with a negative exponent cannot be classified as a polynomial therefore, $\dfrac{{3x}}{{5{\text{y}}}}$ is not a monomial, binomial, trinomial, or polynomial.

Note: Before solving this question, we must know about polynomials and polynomials in one variable. Polynomials: Polynomials are algebraic expressions that comprise of exponents which are added, subtracted or multiplied. When there is only a single variable in the polynomial expression, then that polynomial is called a polynomial in one variable. The terms of polynomials are the parts of the equation which are generally separated by or signs. So, each part of a polynomial in an equation is a term. For example, in a polynomial, say, $2{x^2}\; + {\text{ }}5{\text{ }} + 4$, the number of terms will be $3$. The classification of a polynomial is done based on the number of terms in it.
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