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प्रश्न
If x3 + ax2 − bx+ 10 is divisible by x2 − 3x + 2, find the values of a and b.
उत्तर
Let f(x) = x3 + ax2 − bx + 10 and g(x) = x2 − 3x + 2 be the given polynomials.
We have g(x) = x2 − 3x + 2 = (x − 2) (x − 1)
Clearly, (x − 1) and (x − 2) are factors of g(x)
Given that f(x) is divisible by g(x)
g(x) is a factor of f(x)
(x − 2) and (x − 1) are factors of f(x)
From factor theorem
f(x − 1) and (x − 2) are factors of f(x) then f(1) = 0 and f(2) = 0 respectively.
f(1) = 0
(1)3 + a(1)2 − b(1) + 10 = 0
1 + a − b + 10 = 0
a − b + 11 = 0 ...(i)
f(2) = 0
(2)3 + a(2)2 − b(2) + 10 = 0
8 + 4a − 2b + 10 = 0
4a − 2b + 18 = 0
2(2a − b + 9) = 0
2a − b + 9 = 0 ...(ii)
Subtract (i) from (ii), we get
2a − b + 9 −(a − b + 11) = 0
2a − b + 9 − a + b − 11 = 0
a − 2 = 0
Putting value of a in (i), we get
2 − b + 11 = 0
b = 13
Hence,
a = 2 and b = 13
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