Question

Which of the following is a linear polynomial?
A) $p\left( x \right)=58$
B) $p\left( x \right)=65$
C) $p\left( x \right)=43$
D) $p\left( x \right)=58+65+43+x$

Answer 1

Hint:
Here we have to choose the option which is a linear polynomial. For that, we will use the definition and property of a linear polynomial. Then we will check all the options one by one and we will select the one which will satisfy the property of a linear polynomial.

Complete step by step solution:
A linear polynomial is a polynomial whose degree is one i.e. the highest exponent of the variable is one. A linear polynomial can have at most two terms. In general, a linear polynomial in one variable is of the form:
$p\left( x \right)=ax+b$, where $a$ can’t be zero because if $a$ becomes zero then it will become a constant polynomial.
We can see that the polynomials in first three options are constant polynomials as there is no variable term present there and also the degree of polynomial is zero there i.e. polynomials in first three options are not satisfying the properties of the linear polynomial.
In last option we have,
$\Rightarrow p\left( x \right)=58+65+43+x$
On adding the like terms, we get
$\Rightarrow p\left( x \right)=166+x$
This is of the form $p\left( x \right)=ax+b$, which is a linear polynomial in one variable as there is only one variable present in the polynomial and also the degree of polynomial is one here.
It is satisfying all the properties of a linear polynomial in one variable.
Therefore, $p\left( x \right)=58+65+43+x$ is a linear polynomial.

Thus, the correct option is option D.

Note:
To solve such types of problems, we need to remember the meaning, definition and property of linear polynomials. Also keep in mind that when we draw a graph corresponding to the linear polynomial, we will get a straight line, that is why it is named as linear polynomial.