Question

Classify the following polynomials as polynomials in one variable, two variables, etc.:

1. $x^2+x+1$
2. $y^2-5y$
3. $xy+yz+zx$
4. $x^2-2xy+y^2+1$

Answer 1

Step 1. Check whether the given expressions are polynomial or not

(i). The expression $x^2+x+1$ can be written as $x^2+x^1+x^0$. Here, the exponents of the variable $x$ are only whole numbers (i.e., $2$, $1$ and $0$).

Hence, $x^2+x+1$ is a polynomial.

(ii). The expression $y^2-5y$ can be written as $y^2-5y^1$. Here, the exponents of the variable $y$ are only whole numbers (i.e., $2$ and $1$).

Hence, $y^2-5y$ is a polynomial.

(iii). The expression $xy+yz+zx$ can be written as $x^1{y}^1+y^1{z}^1+z^1{x}^1$. Here, the exponents of the variables are only whole numbers (i.e., $1$).

Hence, $xy+yz+zx$ is a polynomial.

(iv). The expression $x^2-2xy+y^2+1$ can be written as $x^2-2x^1{y}^1+y^2+x^0$. Here, the exponents of the variables are only whole numbers (i.e., $2$, $1$ and $0$).

Hence, $x^2-2xy+y^2+1$ is a polynomial.

Step 2. Classification of a polynomial on the basis of the number of variables.

(i). $x^2+x+1$

The given polynomial consists of only one variable, i.e., $x$.

Hence, the given algebraic expression is a polynomial in one variable.

(ii). $y^2-5y$

The given polynomial consists of only one variable, i.e., $y$.

Hence, the given algebraic expression is a polynomial in one variable.

(iii). $xy+yz+zx$

The given polynomial consists of three variables, i.e., $x,y\mathrm{and}\mathit{}z$.

Hence, the given algebraic expression is a polynomial in three variables.

(iv). $x^2-2xy+y^2+1$

The given polynomial consists of two variables, i.e., $x\mathrm{and}\mathit{}y$.

Hence, the given algebraic expression is a polynomial in two variables.

Therefore, (i) and (ii) are polynomials in one variable, (iii) is a polynomial in three variables, and (iv) is a polynomial in two variables.