Question

Which of the following is a linear polynomial?
\[
  A.{\text{ }}x + {x^2} \\
  B.{\text{ }}x + 1 \\
  C.{\text{ }}5{x^2} - x + 3 \\
  D.{\text{ }}x + \dfrac{1}{x} \\
 \]

Answer 1

Hint- In order to solve the problem use the definition and property of linear polynomials. Check out for all the options given in the problem and select the one which satisfies all the properties.

Complete step-by-step answer:
A linear polynomial is a polynomial of degree one, i.e., the highest exponent of the variable is one.
We note that a linear polynomial in one variable can have at the most two terms. In general, a linear polynomial in one variable will be of the form:
$p\left( x \right) = ax + b,a \ne 0$
The constraint that “a” should not be equal to 0 is required because if “a” is 0, then this becomes a constant polynomial.
As the option A and option C has the degree 2 because they both have the terms containing ${x^2}$
So they cannot be a linear polynomial.
Let us simplify option D
$
  D.{\text{ }}x + \dfrac{1}{x} \\
   \Rightarrow x + \dfrac{1}{x} = \dfrac{{{x^2} + 1}}{x} \\
 $
As after modification of option D we get terms containing ${x^2}$ and also it cannot be converted in general form of linear polynomial, so option D cannot be a linear polynomial.
Let us check option C.
C. $x + 1$
As option c or $x + 1$ is of the form of $p\left( x \right) = ax + b,a \ne 0$
And also has terms with degree 1.
Hence, $x + 1$ is a linear polynomial.
So, option B is the correct option.

Note- In order to solve such problems students must remember the basic definition of linear polynomial and its properties. When we draw the graph corresponding to a linear polynomial, we will get a straight line – hence the name linear.