Question

Which of the following is a linear expression?
A. ${{x}^{2}}+3$
B. ${{y}^{2}}+x$
C. 3
D. 2 + x

Answer 1

Hint: 'Linear' means 'Along a straight line.'
If the graph of an expression is a straight line, then it is a linear expression.
For a linear expression, the degree of the expression must be 1.
In order to find the degree of an expression/equation, the variables must be freed of radicals and rational forms.

Complete step-by-step answer:
The degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients.
For a linear expression, its degree must be 1.
Let's check the degrees of the given expressions one by one:
A. ${{x}^{2}}+3$ : The term ${{x}^{2}}$ has degree 2, which is the highest in the expression. Therefore, the degree of the expression is also 2. Hence, it is not a linear expression.
B. ${{y}^{2}}+x$ : The term ${{y}^{2}}$ has degree 2, therefore the degree of the expression is also 2. Hence, it is also not a linear expression.
C. 3: The constant term has degree 0, therefore the degree of the expression is also 0. Hence, it is not a linear expression.
D. 2 + x: The term x has degree 1, which is the highest in the expression. Therefore, the degree of the expression is also 1. Hence, it is a linear expression.
∴ The correct answer is D. 2 + x.

Note: Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used.
The names for the degrees may be applied to the polynomial or to its terms.
For example, the term 2x in ${{x}^{2}}+2x+3$ is a linear term in a quadratic polynomial.
A polynomial of degree zero is a constant polynomial, or simply a constant.
For a linear polynomial y = f(x), the ratio $\dfrac{\Delta y}{\Delta x}$ is always a constant.