Question

If the zeroes of the polynomial $ f(x) = {x^3} - 12{x^2} + 39x + k $ are in A.P, then the value of k
A. $ 28 $
B. $- 28 $
C. $ 30 $
D. $- 30 $

Answer 1

Hint: For solving the above question, we need to know about the properties of a polynomial and for the AP. For the numbers to be in AP, the consecutive numbers should have the same difference. So let the numbers are $ a - d,a,a + d $ While the polynomial should have the sum of the zeroes is given by the negative of the ratio of the coefficient of the $ {x^2} $ to the coefficient of the $ {x^3} $
the sum of product of the consecutive zeros is given by the ratio of the coefficient of the x to the coefficient of $ {x^3} $
the products of the zeroes are given by negative of the ratio of the coefficient of the constant term to the coefficient of the $ {x^3} $

Complete step by step solution:
In the question, we are given that the zeroes of the polynomial are in AP. So, firstly consider the zeroes of the given polynomial are $ a - d,a,a + d $ (where d is the common difference and a is the constant) as the zeroes are in AP.
Now, on reviving the concepts for the zeroes of the polynomial we got to know that the sum of the zeroes is given by the negative of the ratio of the coefficient of the $ {x^2} $ to the coefficient of the $ {x^3} $
So, $ a - d + a + a + d = \dfrac{{ - ( - 12)}}{1} = 12 $
On solving,
 $
  3a = 12 \\
  a = \dfrac{{12}}{3} = 4 \;
  $
Now, the sum of product of the consecutive zeros is given by the ratio of the coefficient of the x to the coefficient of $ {x^3} $
 $ (a - d)a + a(a + d) + (a - d)(a + d) = \dfrac{{39}}{1} $
On solving,
 $
  {a^2} - da + {a^2} + ad + {a^2} - {d^2} = 39 \\
  3{a^2} - {d^2} = 39 \;
  $
Using the value of a,
 $
  3{\left( 4 \right)^2} - {d^2} = 39 \\
  {d^2} = 48 - 39 = 9 \\
  d = \pm 3 \;
  $
Therefore, the zeroes (or roots) of the polynomial are $ 1,4,7 $ or $ 7,4,1 $
But in the question, we are asked to calculate the value of k
Now, the products of the zeroes are given by negative of the ratio of the coefficient of the constant term to the coefficient of the $ {x^3} $
 $ (a - d)(a)(a + d) = \dfrac{{ - k}}{1} $
Putting the values,
 $
  1 \times 4 \times 7 = - k \\
   \Rightarrow k = - 28 \;
  $
So, the correct option is B.
So, the correct answer is “Option B”.

Note: While solving the mathematical calculations, be careful that taking any number from left-hand side to right-hand side signs always changes. One wrong sign can lead to wrong answers. Formulas of sum and product of roots with respect to the coefficients is very important.