Question

The root of the reciprocal equation of first type and of odd degree is:
A. ${\text{x = 1}}$
B. ${\text{x = - 1}}$
C. ${\text{x = \pm 1}}$
D. ${\text{x = 0}}$

Answer 1

Hint: Reciprocal equations of first type are equations having coefficients from one end of equation are equal in magnitude and sign to the coefficient from the other end. So, we can take a polynomial with odd degree of the form ${\text{a}}{{\text{x}}^{\text{5}}}{\text{ + b}}{{\text{x}}^{\text{4}}}{\text{ + c}}{{\text{x}}^{\text{3}}}{\text{ + c}}{{\text{x}}^{\text{2}}}{\text{ + bx + a = 0}}$such that the magnitude and sign of the coefficients are same from both ends of the equation. Then we can check which of the given solutions satisfy the equation by trial and error method.

Complete step by step Answer:

Reciprocal equations are equations in which the reciprocal of every root of the equation is also its root.
Let ${\text{a}}{{\text{x}}^{\text{5}}}{\text{ + b}}{{\text{x}}^{\text{4}}}{\text{ + c}}{{\text{x}}^{\text{3}}}{\text{ + c}}{{\text{x}}^{\text{2}}}{\text{ + bx + a = 0}}$ be a reciprocal equation of first type and of odd degree. We can find the root by trial and error.
For\[{\text{x = 1}}\], the equation does not satisfy the condition.
For\[{\text{x = 0}}\], the equation does not become 0.
For \[{\text{x = - 1}}\], the equation becomes,
$
  {\text{a}}{\left( {{\text{ - 1}}} \right)^{\text{5}}}{\text{ + b}}{\left( {{\text{ - 1}}} \right)^{\text{4}}}{\text{ + c}}{\left( {{\text{ - 1}}} \right)^{\text{3}}}{\text{ + c}}{\left( {{\text{ - 1}}} \right)^{\text{2}}}{\text{ + b}}\left( {{\text{ - 1}}} \right){\text{ + a}} \\
  {\text{ = - a + b - c + c - b + a = 0}} \\
 $
Therefore, \[{\text{x = - 1}}\] is the root of reciprocal equation of first type and odd degree.
Hence, the correct answer is option B.

Note: Equations of odd degree are equations whose highest power is an odd number. Similarly, equations of even degree are equations whose highest power is an even number. We must not take the coefficients of the reciprocal equation as a, b, c … Instead, we must fix the coefficient based on the condition that coefficients from both ends are equal. Reciprocal equations of the first type are equations having coefficients from one end of the equation are equal in magnitude and sign to the coefficient from the other end. Reciprocal equations of the second type are equations having coefficients from one end of the equation are equal in magnitude and opposite in sign to the coefficient from the other end. The root of an equation is the value of the variable at which the value of the equation becomes zero.
Root of the reciprocal equation of 1st type and of odd degree is \[{\text{x = - 1}}\].
Root of the reciprocal equation of second type and of odd degree is \[{\text{x = 1}}\].
Roots of the reciprocal equation of second type and of even degree is \[{\text{x =+ or - 1}}\].