+647 Find an ordered pair of constants (a,b) such that the polynomial f(x)=x^3+ax^2+(b+2)x+1 is divisible by x^2 - 1.
+129852 If this is divisible by x^2 - 1.....it must be divisible by (x - 1) and ( x + 1)
So....using synthetic division
1 [ 1 a b + 2 1 ]
1 a + 1 a+ b + 3
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1 a + 1 a + b + 3 a + b + 4
-1 [ 1 a b + 2 1 ]
-1 -a + 1 a - b - 3
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1 a - 1 b - a + 3 a - b - 2
And for 1 and -1 to be roots.....the following system must be true
a + b + 4 = 0
a - b - 2 = 0 add these
2a + 2 = 0 ⇒ a = -1
And a + b + 4 = 0
-1 + b + 4 = 0
b + 3 = 0
b = -3
So (a,b) = ( -1, - 3)
