Question

Which of the following is a binomial?
This question has multiple correct options.
A. \[x+2\]
B. \[{{x}^{2}}+2x+3\]
C. \[4{{x}^{2}}\]
D. \[{{x}^{2}}+8\]

Answer 1

Hint: The concept of algebraic expressions is used in this question. Algebraic Expressions can be monomial, binomial, or trinomial. Only one term exists in monomials, two terms exist in binomials, and three terms exist in trinomials.

Complete step by step answer:
Before solving such types of problems, we need to understand the concept of Monomial, Binomial and trinomial.
Monomial: Monomials are algebraic expressions that have only one term, hence the name. To put it another way, it's an expression that comprises any number of similar terms. For example, \[2x+5x+10x\] is a monomial because when it adds the like term it results in \[17x\].
Binomial: A binomial expression is an algebraic expression that only has two terms. A polynomial with only two terms is known as a binomial. For example, \[3{{x}^{2}}+2x\] is a binomial since it contains two unlike terms that is, \[3{{x}^{2}}\] and \[2x\].
Trinomial: Trinomials are algebraic expressions that have three dissimilar terms, hence the name. For example, \[3x+5{{x}^{2}}-6{{x}^{3}}\] is a Trinomial since it contains three unlike terms that is, \[3{{x}}\],\[5{{x}^{2}}\] and \[6{{x}^{3}}\].
Then we come to the problem,
We have to select the binomial in the following options so we take one by one option.
First option, that is \[x+2\]
Here, above the first option we can see that there are two terms.
Hence, \[x+2\] is a binomial.
Now, we have to see the second option that is \[{{x}^{2}}+2x+3\]
Here, we can see that in this equation there are three terms.
Hence, \[{{x}^{2}}+2x+3\] is a trinomial.
Then further we check the third option that is \[4{{x}^{2}}\]
If you notice this then there is only one term.
Hence, \[4{{x}^{2}}\] is a monomial.
Now, we come to the last one that is fourth option which is given as \[{{x}^{2}}+8\]
Here, also if you observe then there are two terms.
Hence, \[{{x}^{2}}+8\] is a binomial.

So, the correct answer is “Option A and D”.

Note:
When we are confronted with such issues, we refer to a few key points. First, we solve each of the following options one by one, then we look for the option with the fewest terms (binomial). As a result, we can quickly choose the best solution.