Fundamental Concepts of Algebra — Statement I-II Type Questions

Statement 1: The algebraic expression $2 + 3x - 5x^2 + \dfrac{4}{x}$ is a polynomial in $x$.

Statement 2: An algebraic expression is said to be a polynomial if the degree of each of its term is a whole number.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

$\dfrac{4}{x} = 4x^{-1}$ has degree -1, which is not a whole number. So the given expression is not a polynomial and Statement 1 is false.

Statement 2 is true, because an algebraic expression is a polynomial only when the exponent of the variable in each of its terms is a whole number (0, 1, 2, 3, .....), i.e. the degree of every term is a whole number. A term with a negative or fractional exponent, such as $\dfrac{4}{x} = 4x^{-1}\text{ or }\sqrt{x} = x^{\frac{1}{2}}$, is not allowed in a polynomial.

Hence, option 4 is the correct option.

Statement 1: The algebraic expression $2x^2 - x + 5x^{-2} + 3x^0 (x \neq 0)$ has no constant term.

Statement 2: In an algebraic expression containing only one variable, the term or terms which is/are independent of the variable is called the constant term.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Since $3x^0 = 3$ (as $x \neq 0$), the given expression does have a constant term 3. So, Statement 1 is false.

Statement 2 is true, because in an algebraic expression having only one variable, a term whose value does not change with the value of the variable, i.e. a term free of the variable, is called the constant term. Here $3x^0 = 3$ is free of $x$ and so 3 is the constant term of the given expression.

Hence, option 4 is the correct option.