Question

x⁴ + 10x³ + 35x² + 50x + 24

Answer 1

Let  \[f\left( x \right) = x^4 + 10 x^3 + 35 x^2 + 50x + 24\]

Now, putting  x = 1,we get

`f(-1) = (-1)^4 + 10(-1)^3 + 35 (-1) +24`

      ` = 1 - 10 + 35 - 50 + 24 = 60 -60`

     ` = 0`

Therefore, (x +1)is a factor of polyno^2 + 50 (-1)mial f(x).

Now,

`f(x) = x^3 (x+1)9x^2(x +1) + 26x(x+1) + 24(x + 1)`

` = (x +1){x^3 +9x^2+ 26x + 24}`

`= (x +1)g(x)      ..... (1)`

Where `g(x) = x^3 + 9x^2 + 26x +24`

Putting x = -2we get

`g(-2) = (-2)^3 + 9(-2)^2 +26 (-2)+ 24`

` = -8 + 36 - 52 + 24 = 60 -60`

` = 0`

Therefore, (x+2)is the factor of g(x).

Now,

`g(x) = x^2 (x+2) + 7x(x + 2) + 12(x + 2)`

` = (x + 2){x^2 + 7x + 12}`

`= (x +2)(x^2 + 4x + 3x + 12)`

` = (x + 2)(x+3)(x + 4)           ........... (2)`

From equation (i) and (ii), we get

 f(x) = (x + 1) (x + 2)(x+3)(x +4)

Hence  (x + 1),(x + 2), (x + 3)  and (x + 4 ) are the factors of polynomial f(x).