If one zero of the polynomial (a2 + 2 ) x2 + 10 x +3a is the reciprocal of the other. Find a.
The zero of a polynomial is a value of the variable in the polynomial that makes the whole polynomial equal to zero. The Sum of the zeroes is -b / a and the product of zeroes is c / a.
Answer: If one zero of the polynomial (a2 + 2 ) x2 + 10 x +3a is the reciprocal of the other. The value of a is 1, 2.
Let's find the value of a.
Given that one zero of the polynomial is reciprocal of the other.
Let one of the zeros be α
The other zero will be 1/α (reciprocal of the first zero)
As we know that, for a given polynomial ax2 + bx + c = 0,
The product of zeroes = c / a, where c is the constant term a is the coefficient of x2 .
As per the question,
(a2 + 2 ) x2 + 10 x + 3a
Constant term = 3a, Coefficient of x2 = (a2 + 2 )
⇒ α × 1/ α = 3a / (a2 + 2 )
By solving the LHS and cross multiplying, we get
⇒ (a2 + 2 ) = 3a
⇒ a2 - 3a + 2 = 0
⇒ a2 - 2a - a + 2 = 0 (By splitting the middle term)
⇒ a (a - 2) -1 (a - 2) = 0
⇒ (a - 2) (a - 1) = 0
Thus, we have two values of a,
⇒ a = 2 and a = 1
Thus, the values of a are 1, 2 for the polynomial (a2 + 2 ) x2 + 10 x +3a.