1. ## Algebra Help

Cubic Equations

Q -
(i) Find the three linear factors of x³ + 5x² - 2x - 24.
(ii) Hence, or otherwise, solve the equation x³ + 5x² - 2x - 24 = 0

Please supply a formula aswell so I can solve these in the future! Thankyou

2. Originally Posted by emrldz
Q -
(i) Find the three linear factors of x³ + 5x² - 2x - 24.
(ii) Hence, or otherwise, solve the equation x³ + 5x² - 2x - 24 = 0

Please supply a formula aswell so I can solve these in the future! Thankyou

Hello,

the good news first: There exists a formula to solve cubic equation: Cardano's formula. It is very complicated and therefore it is very easy to make a lot of silly mistakes.

I can only sho you how I would do your problem:
1. The question indicates that there are rational solutions.
2. Try to find one solution. It must be a factor of 24. I've found x = 2 is a solution because:
$2^3+5 \cdot 2^2 - 2 \cdot 2 - 24 = 8+20-4-24 = 0$
3. Now use synthetic division:
Code:
      |  1    5     -2    -24          <---all coefficients
|  0    2     14     24
------|-------------------------
x = 2 |  1   7      12      0         <---next guess: x = -3
|  0   -3     -12
------|-------------------------
x = -3| 1    4      0
From the last row you know $x + 4 = 0~\iff~x = -4$

Therefore: $x^3+5x^2-2x-24 = (x-2)(x+3)(x+4)$

A product equals zero if one of the factors equal zero:

$x^3+5x^2-2x-24 = =~\iff~x-2=0~\vee~x+3=0~\vee~x+4=0$