Question

if x-2 is a factor of x3 +ax2 + bx+ 16 and a-b=16, find the value of a and b

Answer 1

$x - 2 = 0 \\ x = 2 \\ then \\ x {}^{3 \: } + ax {}^{2} + bx + 16 = 0 \\ 2 {}^{3} + a \times 2 {}^{2} \: + b \times 2 + 16 = 0 \\ 8 + 4a + 2b + 16 = 0 \\ 4a + 2b + 24 = 0 \\2 (2a + b + 16) = 0 \\ 2a + b + 16 = 0 \\ 2a + b = - 16..........(1) \\ and \: we \: know \: that \\ a - b = 16 ..............(2) \\ \\ from \: equation \:1and \: 2 \\ 2a + b \: = - 16 \\ \frac{a \: \: - b \: = \: \: 16}{3a - 0 = 0 } \\ then \\ 3a = 0 \\ a = 0 \\ now \: put \: the \: valueof \: a \: in \: eq.(1) \\ 2a + b = - 16 \\ 2 \times 0 + b = - 16 \\ 0 + b = - 16 \\ b = - 16$
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