Question

Show that (x + 4) , (x − 3) and (x − 7) are factors of x³ − 6x² − 19x + 84

Answer 1

Let  f(x) = x³ − 6x² − 19x + 84 be the given polynomial.

By the factor theorem,

 (x+ 4),(x-3)and  (x-7) are the factor of f(x).

If  f( - 4),f(3)and f(7) are all equal to zero.

Therefore,

`f(-4) = (-4)^3 -6(-4)^2 --9(-4) + 84`

             `= -64 - 96 + 76 + 84`

             ` = -160 + 160`

             ` = 0`

Also

`f(3) = (3)^3 - 6(3)^2 - 19(3) + 84`

` = 27 - 54 - 57 + 84`

` = 111 - 111`

` = 0`

And

`f(7) = (7)^3 - 6(7)^2 - 19(7)+ 84`

`243 - 294 - 133 + 84`

` = 427 - 427`

` =0 `

Hence,  (x + 4),( x - 3)and (x - 7)are the factor of the polynomial f(x).