Question

x⁴ − 7x³ + 9x² + 7x − 10

Answer 1

Let `f(x) = x^4 - 7x^2 + 9x^2 + 7x -10`be the given polynomial.

Now, putting x = 1,we get

\[f(1) = \left( 1 \right)^4 - 7 \left( 1 \right)^3 + 9 \left( 1 \right)^2 + 7\left( 1 \right) - 10\]

\[ = 1 - 7 + 9 + 7 - 10 = 0\]

Therefore, (x-1)is a factor of polynomial f(x).

Now,

\[f(x) = x^4 - x^3 - 6 x^3 + 6 x^2 + 3 x^2 - 3x + 10x - 10\]

`f(x) = x^3 (x-1) - 6x^2 (x-1) + 3x(x-1) + 10(x-1)`

`= (x - 1) {x^3 - 6x^2 +3x + 10}`

` = (x - 1)g(x)          ........ (1)`

Where `g(x) = x^3 - 6x^2 + 32` +10

Putting x = -1we get

`g-(-1) = (-1)^3 -6(-1)^2 + 3(-1) + 10`

` = -1-6 -3 + 10`

` = -10 + 10 = 0`

Therefore, (x + 1)is a factor of g(x).

Now,

\[g(x) = x^3 - 7 x^2 + x^2 - 7x + 10x + 10\]

`g(x) = x^2 (x+1) -7x (x+1)+ 10(x + 1)`

        ` = (x +1){x^2 - 7x + 10}`

        ` = (x+1){x^2 - 5x - 2x + 10}`

        ` = (x+1)(x-2)(x-5)     ........ (2)`

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

 

 f(x) = (x-1)(x+1)(x-2)(x-5)

Hence (x+1),(x-1)(x-2)(x-5) and (x-5) are the factors of polynomial f(x).