Question

If 1, -2 & 3 are roots of x3 -2x2 +ax+6=0 then find ‘a’

Answer 1

EXPLANATION.

1, -2 and 3 are roots of cubic equation.

⇒ x³ - 2x² + ax + 6 = 0.

As we know that,

Let, α, β and γ are the roots of cubic equation.

Sum of the zeroes of the cubic polynomial.

⇒ α + β + γ = - b/a.

Products of the zeroes of the cubic polynomial two at the time.

⇒ αβ + βγ + γα = c/a.

Products of the zeroes of the cubic polynomial.

⇒ αβγ = - d/a.

Now, 1, -2 and 3 are the roots of the cubic polynomial.

Sum of the zeroes of the cubic polynomial.

⇒ 1 + (-2) + 3.

⇒ 4 - 2 = 2.

⇒ 2 = - b/a.

Products of the zeroes of the cubic polynomial two at the time.

⇒ (1)(-2) + (-2)(3) + (3)(1).

⇒ - 2 - 6 + 3.

⇒ - 8 + 3 = - 5.

⇒ - 5 = c/a.

Products of the zeroes of the cubic polynomial.

⇒ (1)(-2)(3).

⇒ - 6 = - d/a.

To find : The value of a.

⇒ - 5 = c/a.

⇒ - 5 = a/1.

⇒ a = - 5.

∴ The value of a is - 5.