Which of the followings is a linear polynomial?

For option (a),

We have x + x²

In this expression, we have 2 terms.

The degree of this expression is 2 (largest exponent of x).

But linear polynomial has a degree of 1, so that it can give exactly one root.

⇒ x + x² is not a linear polynomial.

For option (b),

We have x + 1

In this expression, we have 2 terms.

And the degree of this expression is 1 (largest exponent of x).

This compliments the definition of a polynomial.

⇒ x + 1 is a linear polynomial.

For option (c),

We have 5x² – x + 3.

In this expression, we have 3 terms.

And the degree of this expression is 2 (largest exponent of x).

This violates the definition of linear polynomial, as the degree of linear polynomial is always 1.

⇒ 5x² – x + 3 is not a linear polynomial.

For option (d),

We have

Let us simplify the expression.

⇒

⇒ x² = 1

⇒ x² – 1 = 0

This now is an algebraic expression, having 2 terms.

The degree of this expression is 2 (largest exponent of x).

This again doesn’t satisfy the criteria of linear polynomial, which says that degree should be 1.

⇒ is not a linear polynomial.

Thus, option (b) is correct.