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Question
If x3 − 3x2 + 3x − 7 = (x + 1) (ax2 + bx + c), then a + b + c =
Options
4
12
-10
3
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Solution
The given equation is
x3 − 3x2 + 3x − 7 = (x + 1) (ax2 + bx + c)
This can be written as
\[x^3 - 3 x^2 + 3x - 7 = \left( x + 1 \right)\left( a x^2 + bx + c \right)\]
\[ \Rightarrow x^3 - 3 x^2 + 3x - 7 = a x^3 + b x^2 + cx + a x^2 + bx + c\]
\[ \Rightarrow x^3 - 3 x^2 + 3x - 7 = a x^3 + \left( a + b \right) x^2 + \left( b + c \right)x + c\]
Comparing the coefficients on both sides of the equation.
We get,
a=1 ....... (1)
a+b = -3 ......... (2)
b+c =3 ......... (3)
c = -7 .......(4)
Putting the value of a from (1) in (2)
We get,
1+b =-3
b=-3 -1
b=-4
So the value of a, b and c is 1, – 4 and -7 respectively.
Therefore,
a + b + c =1 - 4 - 7 = -10
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