Answer
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Hint: In this question, we need to find whether the given expressions are polynomial. First let us know about the polynomials. Polynomials are nothing but algebraic expressions that comprise exponents which are added, subtracted or multiplied. By using the fact that if the exponent of the variable is a whole number, the given expression is a polynomial , we can tell that the given expression is a polynomial. Let us check the given expression one by one to check whether it is a polynomial or not.
Complete step-by-step answer:
Here we need to find whether the given expressions are polynomial or not.
First let us check the expression
i. [8\]
This expression can be rewritten as \[8x^{0}\] .
Since the exponent of the variable is a whole number, the given expression is a polynomial.
\[8\] is the polynomial.
ii. \[\sqrt{3}x^{2} – 2x\]
Since the exponent of the variable is a whole number, the given expression is a polynomial.
\[\sqrt{3}x^{2} – 2x\] is a polynomial.
iii . \[1 - \sqrt{5x}\]
This expression can be rewritten as \[1 - \sqrt{5}x^{\dfrac{1}{2}}\ \ \] .
Which is \[1 - \sqrt{5}\left( x \right)^{\dfrac{1}{2}}\]
Since the exponent of the variable is a whole number, the given expression is a polynomial.
\[1 - \sqrt{5x}\] is the polynomial.
iv . \[\dfrac{1}{5x^{- 2}} + 5x + 7\]
This expression can be rewritten as \[\dfrac{1}{5}x^{2} + 5x + 7\] .
Since the exponent of the variable is \[\dfrac{1}{2}\]which is not a whole number, thus the given expression is not a polynomial.
\[\dfrac{1}{5x^{- 2}} + 5x + 7\] is not the polynomial.
v. \[\dfrac{\left( x – 2 \right)\left( x – 4 \right)}{x}\]
Now on simplifying,
We get,
\[\Rightarrow \dfrac{x^{2} – 4x – 2x + 8}{x}\]
On further simplifying,
We get,
\[1 – 6x^{- 1} + 8x^{- 1}\]
Since the exponent of the variable is \[- 1\] which is not a whole number, thus the given expression is not a polynomial.
Therefore \[\dfrac{\left( x – 2 \right)\left( x – 4 \right)}{x}\] is not a polynomial.
vi. \[\dfrac{1}{x + 1}\]
This expression can be rewritten as \[\dfrac{1}{\left( x + 1 \right)^{- 1}}\]
Since the exponent of the variable is not a whole number, the given expression is not a polynomial.
\[\dfrac{1}{x + 1}\] is not the polynomial.
vii . \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\]
Since the exponent of the variable is a whole number, the given expression is a polynomial.
Thus \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\] is a polynomial.
viii. \[\dfrac{1}{2x}\]
This expression can be rewritten as \[\dfrac{1}{2}\left( x \right)^{- 1}\] .
Since the exponent of the variable is not a whole number, the given expression is not a polynomial.
\[\dfrac{1}{2x}\] is not the polynomial.
Therefore (i) \[8\] , (ii) \[\sqrt{3}x^{2} – 2x\] (iv) \[\dfrac{1}{5x^{- 2}}\ + 5x + 7\] and (vii) \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\] are polynomials.
Final answer :
The expressions (i) \[8\] , (ii) \[\sqrt{3}x^{2} – 2x\] (iv) \[\dfrac{1}{5x^{- 2}}\ + 5x + 7\] and (vii) \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\] are polynomials.
Option B). Polynomials: (i), (ii), (iv), (vii)
So, the correct answer is “Option B”.
Note: In order to solve these types of questions, we should have a strong grip over polynomials . We also need to know that polynomials are of different types namely Monomial, Binomial, and Trinomial. A monomial is nothing but a polynomial with one term. For example, \[4x^{2}\] . A binomial is nothing but a polynomial with two unlike terms. For example, \[3xy^{2}-x^{2}\] . A trinomial is nothing but a polynomial with three terms which are unlike. For example, \[5xy + 6x^{2} - 7\] .
Complete step-by-step answer:
Here we need to find whether the given expressions are polynomial or not.
First let us check the expression
i. [8\]
This expression can be rewritten as \[8x^{0}\] .
Since the exponent of the variable is a whole number, the given expression is a polynomial.
\[8\] is the polynomial.
ii. \[\sqrt{3}x^{2} – 2x\]
Since the exponent of the variable is a whole number, the given expression is a polynomial.
\[\sqrt{3}x^{2} – 2x\] is a polynomial.
iii . \[1 - \sqrt{5x}\]
This expression can be rewritten as \[1 - \sqrt{5}x^{\dfrac{1}{2}}\ \ \] .
Which is \[1 - \sqrt{5}\left( x \right)^{\dfrac{1}{2}}\]
Since the exponent of the variable is a whole number, the given expression is a polynomial.
\[1 - \sqrt{5x}\] is the polynomial.
iv . \[\dfrac{1}{5x^{- 2}} + 5x + 7\]
This expression can be rewritten as \[\dfrac{1}{5}x^{2} + 5x + 7\] .
Since the exponent of the variable is \[\dfrac{1}{2}\]which is not a whole number, thus the given expression is not a polynomial.
\[\dfrac{1}{5x^{- 2}} + 5x + 7\] is not the polynomial.
v. \[\dfrac{\left( x – 2 \right)\left( x – 4 \right)}{x}\]
Now on simplifying,
We get,
\[\Rightarrow \dfrac{x^{2} – 4x – 2x + 8}{x}\]
On further simplifying,
We get,
\[1 – 6x^{- 1} + 8x^{- 1}\]
Since the exponent of the variable is \[- 1\] which is not a whole number, thus the given expression is not a polynomial.
Therefore \[\dfrac{\left( x – 2 \right)\left( x – 4 \right)}{x}\] is not a polynomial.
vi. \[\dfrac{1}{x + 1}\]
This expression can be rewritten as \[\dfrac{1}{\left( x + 1 \right)^{- 1}}\]
Since the exponent of the variable is not a whole number, the given expression is not a polynomial.
\[\dfrac{1}{x + 1}\] is not the polynomial.
vii . \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\]
Since the exponent of the variable is a whole number, the given expression is a polynomial.
Thus \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\] is a polynomial.
viii. \[\dfrac{1}{2x}\]
This expression can be rewritten as \[\dfrac{1}{2}\left( x \right)^{- 1}\] .
Since the exponent of the variable is not a whole number, the given expression is not a polynomial.
\[\dfrac{1}{2x}\] is not the polynomial.
Therefore (i) \[8\] , (ii) \[\sqrt{3}x^{2} – 2x\] (iv) \[\dfrac{1}{5x^{- 2}}\ + 5x + 7\] and (vii) \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\] are polynomials.
Final answer :
The expressions (i) \[8\] , (ii) \[\sqrt{3}x^{2} – 2x\] (iv) \[\dfrac{1}{5x^{- 2}}\ + 5x + 7\] and (vii) \[\dfrac{1}{7}a^{3} - \dfrac{1}{\sqrt{3}}a^{2} – 4a + 7\] are polynomials.
Option B). Polynomials: (i), (ii), (iv), (vii)
So, the correct answer is “Option B”.
Note: In order to solve these types of questions, we should have a strong grip over polynomials . We also need to know that polynomials are of different types namely Monomial, Binomial, and Trinomial. A monomial is nothing but a polynomial with one term. For example, \[4x^{2}\] . A binomial is nothing but a polynomial with two unlike terms. For example, \[3xy^{2}-x^{2}\] . A trinomial is nothing but a polynomial with three terms which are unlike. For example, \[5xy + 6x^{2} - 7\] .
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