Sets, Relations & Functions Question - None

Question

Let $$P(x) = kx^3 + 2k^2x^2 + k^3$$ . If $$(x-2)$$ is a factor of $$P(x)$$ , find the sum of all real values of $$k$$ .

Accepted Answer

Key Idea: Use the factor theorem: $$(x-2)$$ is a factor iff $$P(x)$$ .

Step 1: $$$P(2) = 8k + 8k^2 + k^3 = 0$$$

Step 2: Factor out $$k$$ : $$$k(k^2 + 8k + 8) = 0$$$ Real roots: $k=0,\ k=-4\pm 2\sqrt{2}$

Step 3: Sum $= 0 + (-4+2\sqrt{2}) + (-4-2\sqrt{2}) = -8$

Answer: $$-8$$

Trap: Do not treat the condition as

$P(x)=0$

for all

$x$

. Factor theorem requires only the single substitution

$x=2$

.

Key Concept: Factor theorem + Vieta's formulas for sum of roots

Question: