**Question:**

Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category?

(i) *x + y*

(ii) 1000

(iii) *x* + *x*2 + *x*3 + 4*y*4

(iv) 7 + *a* + 5*b*

(v) 2*b* − 3*b*2

(vi) 2*y* − 3*y*2 + 4*y*3

(vii) 5*x* − 4*y* + 3*x*

(viii) 4*a* − 15*a*2

(ix) *xy + yz + zt + tx*

(x) *pqr*

(xi) *p*2*q* + *pq*2

(xii) 2*p* + 2*q*

**Solution:**

Definitions:

A polynomial is monomial if it has exactly one term. It is called binomial if it has exactly two non-zero terms. A polynomial is a trinomial if it has exactly three non-zero terms.

(i) The polynomial $x+y$ has exactly two non zero terms, i.e., $x$ and $y$. Therefore, it is a binomial.

(ii) The polynomial 1000 has exactly one term, i.e., 1000 . Therefore, it is a monomial.

(iii) The polynomial $x+x^{2}+x^{3}+x^{4}$ has exactly four terms, i.e., $x, x^{2}, x^{3}$ and $x^{4}$. Therefore, it doesn't belong to any of the categories.

(iv) The polynomial $7+a+5 b$ has exactly three terms, i.e., $7, a$ and $5 b$. Therefore, it is a trinomial.

(v) The polynomial $2 b-3 b^{2}$ has exactly two terms, i.e., $2 b$ and $-3 b^{2}$. Therefore, it is a binomial.

(vi) The polynomial $2 y-3 y^{2}+4 y^{3}$ has exactly three terms, i.e., $2 y,-3 y^{2}$ and $4 y^{3}$. Therefore, it is a trinomial.

(vii) The polynomial $5 x-4 y+3 x$ has exactly three terms, i.e., $5 x,-4 y$ and $3 x$. Therefore, it is a trinomial.

(viii) The polynomial $4 a-15 a^{2}$ has exactly two terms, i.e., $4 a$ and $-15 a^{2}$. Therefore, it is a binomial.

(ix) The polynomial $x y+y z+z t+t x$ has exactly four terms $x y, y z, z t$ and $t x$. Therefore, it doesn't belong to any of the categories.

(x) The polynomial pqr has exactly one term, i.e., pqr. Therefore, it is a monomial.

(xi) The polynomial $p^{2} q+p q^{2}$ has exactly two terms, i.e., $p^{2} q$ and $p q^{2}$. Therefore, it is a binomial.

(xii) The polynomial $2 p+2 q$ has two terms, i.e., $2 p$ and $2 q$. Therefore, it is a binomial.