Question

Classify the following polynomial as polynomial in one variable, two variables etc.
(a)${{x}^{2}}-xy+7{{y}^{2}}$
(b)${{x}^{2}}-2tx+7{{t}^{2}}-x+t$
(c)${{t}^{3}}-3{{t}^{2}}+4t-5$
(d)$xy+yz+zx$

Answer 1

Hint: First we write the meaning of having polynomials in one variable, in two variables and in three variables etc. And after that we will look at all the options and name as the category in which they belong.

Complete step-by-step answer:
First we will write the definition of polynomial,
Polynomial: In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Now we will write the meaning of one variable etc.
One variable polynomial: The polynomial function which contains only a single variable like x.
Two variable polynomials: The polynomial function which contains two variables like x,y.
Three variable polynomials: The polynomial function which contains three variables like x,y,z.
Now let’s check all the four options and in which category it falls,
For (a): ${{x}^{2}}-xy+7{{y}^{2}}$
It contains two variables x and y.
Hence it is a two variable polynomial.
For (b): ${{x}^{2}}-2tx+7{{t}^{2}}-x+t$
It contains two variables x and t.
Hence it is a two variable polynomial.
For (c): ${{t}^{3}}-3{{t}^{2}}+4t-5$
It contains a single variable t.
Hence it is a one variable polynomial.
For (d): $xy+yz+zx$
It contains three variables x, y and z.
Hence it is a three variable polynomial.

Note: We have used the definition of one, two, three variable polynomials to check all the options. So, one should understand the definitions of all these three terms to solve this question correctly.