Question

If the zeros of the polynomial f(x)=2x3-15x2+37x-30 are in a.p., find them.

Answer 1

2,5/2,3 are roots of given equation

Answer 2

$\huge\fbox \red{Solution:}$

Let the zeros of the given polynomial be α, β and γ. (3 zeros as it’s a cubic polynomial)

And given, the zeros are in A.P.

So, let’s consider the roots as

α = a – d, β = a and γ = a +d

Where, a is the first term and d is the common difference.

From given f(x), a= 2, b= -15, c= 37 and d= 30

⇒ Sum of roots = α + β + γ = (a - d) + a + (a + d) = 3a = (-b/a) = -(-15/2) = 15/2

So, calculating for a, we get 3a = 15/2 ⇒ a = 5/2

⇒ Product of roots = (a - d) x (a) x (a + d) = a(a2 –d2) = -d/a = -(30)/2 = 15

⇒ a(a2 –d2) = 15

Substituting the value of a, we get

⇒ (5/2)[(5/2)2 –d2] = 15