If the zeroes of the cubic polynomial x3 - 6x2 + 3x + 10 are of the fo

1.	If the zeroes of the cubic polynomial x3 - 6x2 + 3x + 10 are of the form a, a + b and a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
Answer» Let P(x) = x³ - 6x² + 3x + 10 And (a), (a + b) and (a + 2b) are the zeroes of P(x). We know, Sum of the zeroes = - (coefficient of x²) ÷ coefficient of x³ α + β + γ = - b/a a + (a + b) + (a + 2b) = - (- 6) 3a + 3b = 6 a + b = 2 ⇒ a = 2 - b (1) Product of all the zeroes = - (constant term) ÷ coefficient of x³ αβγ = - 10 a(a + b)(a + 2b) = - 10 (2 - b) (2) (2 + b) = - 10 (2 - b) (2 + b) = - 5 4 - b² = - 5 ⇒ b² = 9 ⇒ b = +-3 When b = 3, a = 2 - 3 = - 1 (from (1)) ⇒ a = - 1 When b = - 3,a = 2 - (- 3) = 5 (from(1)) ⇒ a = 5 Case 1: when a = - 1 and b = 3 The zeroes of the polynomial are: a = - 1 a + b = - 1 + 3 = 2 a + 2b = - 1 + 2(3) = 5 ⇒ - 1, 2 and 5 are the zeroes Case 2: when a = 5,b = - 3 The zeroes of the polynomial are: a = 5 a + b = 5 - 3 = 2 a + 2b = 5 - 2(3) = - 1 ⇒ - 1, 2 and 5 are the zeroes By both the cases the zeroes of the polynomial are - 1, 2, 5

If the zeroes of the cubic polynomial x3 - 6x2 + 3x + 10 are of the form a, a + b and a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Answer»

Let P(x) = x³ - 6x² + 3x + 10

And (a), (a + b) and (a + 2b) are the zeroes of P(x).

We know,

Sum of the zeroes = - (coefficient of x²) ÷ coefficient of x³

α + β + γ = - b/a

a + (a + b) + (a + 2b) = - (- 6)

3a + 3b = 6

a + b = 2

⇒ a = 2 - b (1)

Product of all the zeroes = - (constant term) ÷ coefficient of x³

αβγ = - 10

a(a + b)(a + 2b) = - 10

(2 - b) (2) (2 + b) = - 10

(2 - b) (2 + b) = - 5

4 - b² = - 5
⇒ b² = 9

⇒ b = +-3

When b = 3, a = 2 - 3 = - 1 (from (1))

⇒ a = - 1

When b = - 3,a = 2 - (- 3) = 5 (from(1))

⇒ a = 5

Case 1: when a = - 1 and b = 3

The zeroes of the polynomial are:

a = - 1 a + b = - 1 + 3 = 2 a + 2b = - 1 + 2(3) = 5

⇒ - 1, 2 and 5 are the zeroes

Case 2: when a = 5,b = - 3

The zeroes of the polynomial are:

a = 5 a + b = 5 - 3 = 2 a + 2b = 5 - 2(3) = - 1

⇒ - 1, 2 and 5 are the zeroes

By both the cases the zeroes of the polynomial are - 1, 2, 5