Question

If α, β and γ are common integral solutions of these inequations, then remainder obtained when p(x) = (x – α)(x β)(x - γ) is divided by (x - 6) is​

Answer 1

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Answer 2

Given:

The solution of the equation (x- α)(x- β)(x-γ) is α, β, and γ

To Find:

We have to find the remainder when (x- α)(x- β)(x-γ) is divided by (x- 6)

Solution:

First Step

We need to simplify the given equation to the general form of equation.

p(x) = (x- α)(x- β)(x-γ)

      = (x²- (α+ β)x + αβ)×(x-γ)

      = x³- (α+ β)x² + αβx- γx²+ (α+ β)γx+ αβγ

      = x³- (α+ β+ γ)x² + (αβ+ βγ+ αγ)x+ αβγ

⇔ p(x) = x³- (α+ β+ γ)x² + (αβ+ βγ+ αγ)x+ αβγ

Second step

We have to divide x³- (α+ β+ γ)x² + (αβ+ βγ+ αγ)x+ αβγ  by (x- 6)

Put Divisor = 0

    ∵   (x- 6) = 0

                x = 6

Third Step

Put x = 6 in p(x) = x³- (α+ β+ γ)x² + (αβ+ βγ+ αγ)x+ αβγ

⇒ p(6) = 6³- (α+ β+ γ)6² + (αβ+ βγ+ αγ)6+ αβγ

           = 216- 36(α+ β+ γ)+ 6(αβ+ βγ+ αγ) + αβγ

Thus, remainder of p(6) = 216- 36(α+ β+ γ)+ 6(αβ+ βγ+ αγ) + αβγ