
If is a root of then the other roots are
A. ,
B. ,
C.
D.
Answer
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Hint: As we know that imaginary roots always occur in the form of conjugate pairs, we will find the conjugate pair and thus multiply both the roots and find the equation. Next, we will substitute in to determine the value of and after getting the value of , we will factorize the equation and get the one root in the form of equation as and find the other factor also. Thus, we can find the remaining roots of the equation using .
Complete step by step solution
As given is a root of the given equation,
We can determine the conjugate pair of .
That is .
Let and , which can be written as and respectively.
Now, we will multiply both the equations obtained above and use wherever needed.
Thus, we get,
Hence, the roots of the equation are and .
Further, we will substitute in to evaluate the value of
Use and in the equation and solve for .
Thus, we get,
Thus, the given equation becomes,
Next, factorize the above equation in the form such that one of the factors is :
Apply zero-product rule on the obtained factors to solve for .
or
Thus, for the factor we know the roots as and .
So now we need to solve the equation to determine the roots by using the formula , where and .
Thus, we get,
Thus, the other roots of the given equation are
Hence, the correct option is (3).
Note: As the given equation is of degree 4 so, we will have 4 roots of the equation. As one imaginary root is given and the conjugate of the given root is also the root of the equation, thus, we have 2 roots and need to find the other 2 roots of the equation. By substituting the value of we can find the value of and after that only we can determine the other 2 roots of the equation. As we can not find the factors using middle term splitting method so, we have used the formula to evaluate the other 2 roots.
Complete step by step solution
As given
We can determine the conjugate pair of
That is
Let
Now, we will multiply both the equations obtained above and use
Thus, we get,
Hence, the roots of the equation
Further, we will substitute
Use
Thus, we get,
Thus, the given equation becomes,
Next, factorize the above equation in the form such that one of the factors is
Apply zero-product rule on the obtained factors to solve for
Thus, for the factor
So now we need to solve the equation
Thus, we get,
Thus, the other roots of the given equation are
Hence, the correct option is (3).
Note: As the given equation is of degree 4 so, we will have 4 roots of the equation. As one imaginary root is given and the conjugate of the given root is also the root of the equation, thus, we have 2 roots and need to find the other 2 roots of the equation. By substituting the value of
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