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**

**PLEASE HELP! VERY VERY URGENT**

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:

$\displaystyle 2^3+5 \cdot 2^2 - 2 \cdot 2 - 24 = 8+20-4-24 = 0$

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 $\displaystyle x + 4 = 0~\iff~x = -4$

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

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

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