Answer 1

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