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.

Explanation:

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.