Question

Assertion :If $a,b,c\in Q$ and $2^{\frac{1}{3}}$ is a root of $a{x}^2+bx+c=0,thena=b=c=0$. Reason: A polynomial equation with rational coefficients cannot have irrational roots

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$a,b,cϵQ$

$2^{1/3}$ is root of $a{x}^2+bx+c=0$

$\therefore a\left(2^{2/3}\right)+b\left(2^{1/3}\right)+C=0\to 1$

a,b,c are rational, $2^{2/3},2^{1/3}$ are irrational

$1$ is true only if $a=b=c=0$

A polynomial of degree 'n' can have irrational roots with power $\ge \frac{1}{n}$ (i.e., $\frac{1}{n^{th}}$ irrational power numbers)

$\therefore$ For a quadratic equation $(a_1^{1/2}+b_1^{1/2}+c_1^{1/2}....+z_1^{1/2})$

is possible as a root but $(a_2^{1/3}+b_2^{1/3}+....+z_2^{1/3})$ is not possible as a root

$(\because \frac{1}{3}<\frac{1}{2})$