The polynomial \(x^6 + ax + b\) is divisible by \(x^2 - 2x - 1.\) Find \(a + b.\)
+37144 Here is my scrawled answer..... basically I did the long division and found what values of A and B produce a zero remainder....
a+b = -99
+129845 x^4 + 2x^3 + 5x^2 + 12x + 29
x^2 - 2x - 1 [ x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + ax + b ]
x^6 - 2x^5 - x^4
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2x^5 + x^4 + 0x^3
2x^5 - 4x^4 - 2x^3
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5x^4 + 2x^3 + 0x^2
5x^4 -10x^3 - 5x^2
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12x^3 + 5x^2 + ax
12x^3 -24x^2 -12x
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29x^2 + ( a + 12)x + b
29x^2 - 58x -29
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(a + 12 + 58)x + (b + 29)
(a + 70)x + (b + 29) = 0
ax + 70x + b + 29 = 0
ax + b = -70x - 29
So
a + b = -70 + (-29) = -99
