Question

2x⁴ − 7x³ − 13x² + 63x − 45

Answer 1

Let  f(x) 2x⁴ − 7x³ − 13x² + 63x − 45 be the given polynomial.

Now, putting x = 1we get

`f(1) = 2(1)^4 - 7(1)^3 - 13(1)^2 + 63(1) - 45`

` = 2-7 -13 + 63 - 45`

` = 65 - 65 = 0`

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

Now,

`f(x) = 2x^3 (x -1) - 5x^2(x -1) - 18x (x-1) + 45(x -1)`

` = (x -1) {2x^3 - 5x^2  - 18x + 45}`

` = (x -1)g(x)          .... (1)`

Where `g(x)  = 2x^2 - 5x^2 - 18x + 45`

Putting x = 3,we get

`g(3) = 2(3)^3 - 5(3)^2 - 18(3) + 45`

       ` = 54 - 45 - 54 + 45`

       ` = 0`

Therefore, (x -3)is a factor of g(x).

Now,

`g(x) = 2x^2 (x -3) + x(x -3) - 15(x -3)`

` = (x -3){2x^2 + x - 15}`

` = (x -3){2x ^2 + 6x - 5x - 15}`

` = (x -3){(2x - 5)(2x +3)}`

` = (x - 3) (x + 3)(2x -5) ..............(2)`

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

`f(x) = (x -1) (x -3)(x +3) (2x -5)`

Hence  (x - 1),(x - 3),(x + 3) and (2x-5)are the factors of polynomial f(x).