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If $\alpha$ is a root of $4x^{2}+2x-1= 0$ then $4\alpha ^{3}-2\alpha +\frac{1}{2}$ equals

Option 1)
$0$

Option 2)
$1$

Option 3)
$2$

Option 4)
$3$

Answers (1)

best_answer

$\because \: \alpha$ is a root of $4x^{2}+2x-1=0$

$\Rightarrow \: 4\alpha ^{2}=1-2\alpha$

$\therefore \: 4\alpha ^{3}-2\alpha +\frac{1}{2}=\alpha \left ( 4\alpha ^{2} \right )-2\alpha +\frac{1}{2}$

$=\: \alpha \left ( 1-2\alpha \right )-2\alpha +\frac{1}{2}$

$=\: -2\alpha ^{2}-\alpha +\frac{1}{2}$

$=\: \frac{-4\alpha ^{2}-2\alpha +1}{2}$

$=\: \frac{-\left ( 1-2\alpha \right )-2\alpha +1}{2}$

$=\: 0$

$\therefore$ Option (A)

Degree reduction -

If $\alpha$ is a root of $ax^{2}+bx+c= 0$ then $a\alpha^{2}+b\alpha+c= 0\Rightarrow a\alpha ^{2}= -b\alpha -c$ which can be used to reduce degree of $\alpha$ from $2$ to $1$ .

-

Option 1)

$0$

This is correct

Option 2)

$1$

This is incorrect

Option 3)

$2$

This is incorrect

Option 4)

$3$

This is incorrect

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prateek

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