A cubic polynomial $x^{3} + p x^{2} + q x + r$ has roots $2$ , $- 3$ , and $k$ . If the sum of the squares of the roots is $17$ , find all possible values of $k$ .

hard 3 marks Class 9 · Mathematics · Polynomials

Answer

k = 2

k = - 2

The correct answer is $k = 2$ or $k = - 2$ — So $k = 2$ or $k = - 2$ .

Step-by-step solution

For a cubic with roots $α, β, γ$ , the sum of squares is $α^{2} + β^{2} + γ^{2} = (α + β + γ)^{2} - 2 (α β + β γ + g ammaa l p ha)$ .
Compute $α + β + γ = 2 + (- 3) + k = k - 1$ .
Compute pairwise sums: $α β + β γ + g ammaa l p ha = 2 (- 3) + (- 3) k + 2 k = - 6 - k$ .
Thus $α^{2} + β^{2} + γ^{2} = (k - 1)^{2} - 2 (- 6 - k) = k^{2} + 13$ .
Set this equal to $17$ : $k^{2} + 13 = 17 im pl i es k^{2} = 4$ .
So $k = 2$ or $k = - 2$ .

Common mistake: Forgetting to square the sum of roots

Concept tested: Sum of squares of roots in a cubic