If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is reciprocal of the other, find the value of a.
A polynomial is defined as an algebraic expression whose maximum power of the variable must be a non-negative integer.
Answer: If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is the reciprocal of the other, then the value of a is 3.
Let us proceed step by step to find the value of a.
According to the question, one of the zeros is the reciprocal of the other, so, let us consider one zero to be x.
Therefore, the other zero will be 1 / x, and the product of zeros will be 1.
For any polynomial of the form ax2+ bx + c = 0,
Sum of zeros = - b / a
Product of zeroes = c / a
Using these results for the equation given in the question (a2 + 9)x2 + 13x + 6a, we get
The product of zeros will be c / a = 6a / (a2+ 9) = 1
⇒ a2+ 9 = 6a
⇒ a2- 6a + 9 = 0 [rearranging terms]
⇒ a2 – 3a – 3a + 9 = 0 [splitting middle term]
⇒ a (a - 3) -3 (a - 3) = 0 [taking a as common in 1st two terms and - 3 as common in last two terms]
⇒ (a - 3) (a - 3) = 0
⇒ a = 3 , 3