Question

Para quais valores de a, b e c o polinômio seria identicamente nulo:

a) P(x) = (a -5)x2 + (2a + b)x

b) P(x) = (a –b)x2 + (a + b – 8)x + (b - 3c + 5)

c) P(x) = (2a – 1)x2 + (3a – 2b)x + (4a – c)

Answer 1

$P(x) = (a-5)x^2+(2a+b)x\\\\a - 5 = 0\\\boxed{a = 5}\\\\2a + b = 0\\2 \cdot 5 + b = 0\\10 + b = 0\\\boxed{b = -10}$

$P(x) = (a-b)x^2 + (a+b-8)x + (b-3c+5)\\\\a - b = 0\\a = b\\\\a + b - 8 = 0\\b + b - 8 = 0\\2b = 8\\b = \frac{8}{2}\\\boxed{b = 4}\\\\a = b\\\boxed{a = 4}\\\\b - 3c + 5 = 0\\4 - 3c + 5 = 0\\-3c + 9 = 0\\c = \frac 93\\\boxed{c = 3}$

$P(x) = (2a-1)x^2+(3a-2b)x+(4a-c)\\\\2a - 1 = 0\\2a = 1\\\boxed{a = \frac 12}\\\\\\3a - 2b = 0\\3 \frac 12 - 2b = 0\\\frac 32 - 2b = 0\\3 - 4b = 0\\4b = 3\\\boxed{b = \frac 34}\\\\\\4a-c = 0\\4\frac 12 - c = 0\\\frac 42 - c =0\\2 - c = 0\\\boxed{c = 2}$