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Theorems for Continuity

If x3+a x2-b ...

Question

If $x^{3} + a x^{2} - b x + 10$ is exactly divisible by $x^{2} + 3 x + 2$ . Find the values of a and b

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Solution

$x^{2} + 3 x + 2 = (x + 2) (x + 1))$
Since f(x) is divisible by $(x + 2) (x + 1)$
$f (- 2) & (f - 1)$ must be zero \rightarrow Remainder theorem
$f (- 2) = (- 2)^{3} + a (- 2)^{2} - b (- 2) + 10 = 0 - 8 + 4 a + 2 b + 10 = 0$
or $4 a + 2 b + 2 = 0$ -----(1)
$f (- 1) = (- 1)^{3} + a (- 1)^{2} - b (- 1) + 10 = 0 = - 1 + a + b + 10 = 0 a + b + 9 = 0$ -----(2)
Solving $(1) & (2)$
put $a = - (b + a) i s (1) - 4 (b + 9) + 2 b + 2 = 0 - 4 b - 36 + 2 b + 2 = 0 - 2 b - 34 = 0 o r b = 17 a = - (17 + 9) = - 26$

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