Question

Classify the following polynomials as monomials, binomials, and trinomials. Which polynomials do not fit in any of these three categories?
\[x + y\]
\[1000\]
\[x + {x^2} + {x^3} + {x^4}\]
\[7 + y + 5x\]
\[2y - 3{y^2}\]
\[2y - 3{y^2} + 4{y^3}\]
\[5x - 4y + 3xy\]
\[4z - 15{z^2}\]
\[ab + bc + cd + da\]
\[pqr\]
\[{p^2}q + p{q^2}\]
\[2p + 2q\]

Answer 1

Hint: A polynomial is an expression that consists of variables and coefficients, and it involves the operations as addition, subtraction, multiplication, and non-negative exponentiation of the variables. \[a{x^2} + bx + c\]is a polynomial as it contains variables, their coefficients and involves the operation of addition. There are three basic types of polynomials, namely Monomial, Binomial, and Trinomial.

Complete step-by-step answer:
A monomial is a polynomial consisting of only one term; it includes the number, whole number, or variables that are multiplied together. A monomial can be a constant, a variable, or a multiple of numbers and variables. \[Eg. - x,1,xy\]
A binomial is a polynomial consisting of two terms or two monomials. A binomial can be the sum of two constants, variables, or two multiples of numbers and variables. \[Eg. - x + y,1 + x,xy + z\]
A trinomial is a polynomial consisting of three terms or three monomials. A trinomial can be the sum of three constant or three variables or three multiples of numbers and variables. \[Eg. - x + y + z,1 + x + z,xy + z + 5,a + b + c\]
Here, in the question we need to count the numbers of terms in each of the given expressions and then classify them in categories of polynomials as discussed above.
Categorize each polynomial by counting the terms of the number in each given expressions,\[x + y\],\[1000\],\[x + {x^2} + {x^3} + {x^4}\], \[7 + y + 5x\],\[2y - 3{y^2}\], \[2y - 3{y^2} + 4{y^3}\],\[5x - 4y + 3xy\],\[4z - 15{z^2}\], \[ab + bc + cd + da\],\[pqr\], \[{p^2}q + p{q^2}\], \[2p + 2q\]
Monomials (Expressions with one term) - \[1000\],\[pqr\]
Binomials (Expressions with two terms) - \[x + y\],\[2y - 3{y^2}\],\[4z - 15{z^2}\], \[{p^2}q + p{q^2}\],\[2p + 2q\]
Trinomials (Expressions with three terms) -\[7 + y + 5x\], \[2y - 3{y^2} + 4{y^3}\],\[5x - 4y + 3xy\]
Now we are left with the polynomial \[x + {x^2} + {x^3} + {x^4}\], and since it contains 4 terms hence, it cannot be classified as any polynomial.

Note: While determining the polynomials, students must be careful while counting the numbers of terms because the categorization is purely dependent upon the number of terms.