Synthetic Division/ Polynomial Help.........

Use synthetic division to show that, x = - 5

is the root of the equation, x^3 - 4x^2 - 27x + 90 = 0

Then, use the result to factor the polynomial completely into the form, (x + A) (x + B) (x + C)

Then give the list of values A,B,C that you get.

Thanks for your help!

substitute - 5 in the left side of the equation
(-5)^3 - 4(-5)^2 - 27(-5) + 90 = -125 - 4 (25) +135 + 90 = 0

1 - 4 - 27 +90

-5 -5 +45 -90

| 1 -9 +18 | 0

Therefore x^3 - 4x^2 - 27x + 90 = (x + 5)(x^2 - 9x +18)

Apply a similar process with x = 3

1 - 9 18

3 3 -18

1 - 6 | 0

so x^3 - 4x^2 - 27x + 90 = (x + 5)(x - 3)(x - 6)

the values you get are
5
- 3
- 6

The synthetic form of the cubic polynomial is 1, -4, -27, 90

Synthetic division yields the following

.......... 1 ..... -4 ..... -27 ..... 90
.................. -5 ....... 45 ..... -90
--------------------------------------…
...-5) .. 1 ..... -9 ..... 18 ..... 0 this is the remainder

Due to the remainder theorem, the value of the polynomial for x = -5 is 0 a root of the polynomial

Then x - -5 is a factor, the other factor is 1x^2 + -9x + 18 = (x - 6)(x - 3)

The other solutions are x = 6 and x = 3

Check these results in the original equation, I did!

-5 . . |||| . .1. . . . -4. . . . . .-27. . . . .90
. . . . . . . . . . . . -5. . . . . . 45. . . . .-90
- - - - - - - - - - - - - - - - - - - - - - - - - - - - (+)
. . . . . . . 1. . . . .-9. . . . . .18. . . . . 0



x^3 - 4x^2 - 27x + 90 = 0

(x^2 - 9x + 18)(x + 5) = 0

(x - 6)(x - 3)(x + 5) = 0

x = -5

or

x = 3

or

x = 6

-5I1.....- 4......-27 ........90
...........-5........45.........-90
_______________________
... 1......-9.......18..........0


(x^3 - 4x^2 - 27x + 90) = (x + 5)(x^2 - 9x + 18)

=> (x + 5)(x - 6)(x - 3) = 0

A = 5, B = 6 and C = 3