Question

If the zeros of the polynomial f(x) = x³ − 12x² + 39x + k are in A.P., find the value of k.

Answer 1

Let a - d, a and a + d be the zeros of the polynomial f(x). Then,

Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`

`a-d+a+a+d=(-(-12))/1`

`a-d+a+a+d=12`

`3a=12`

`a=12/3=4`

Since 'a' is a zero of the polynomial f(x).

f(x) = x³ − 12x² + 39x + k

f(a) = 0

f(a) = 4³ − 12(4)² + 39(4) + k

0 = 64 - 192 + 156 + k

0 = 220 - 192 + k

0 = 28 + k

-28 = k

Hence, the value of k is -28.