Using factor theorem, factorize the polynomial `x^4-2x^3-13 x^2+14 x+24`

Explanation

`x^4-2x^3-13 x^2+14 x+24`
Firstly, we will find the root of `p(x) = x^4 + 2x³ - 13x² - 14x + 24`
When `x = 1`
`= x^4 + 2x³ - 13x² - 14x + 24`
`= (1)^4 + 2(1)³ - 13(1)² - 14(1) + 24`
`= 1 + 2 - 13 - 14 + 24`
`= 27 - 27`
`= 0`
So by factor theorem, `(x - 1)` is a factor
On dividing `p(x)` by `(x - 1)`, we get `x³ + 3x² - 10x -24`
Now find the root of `p(x) = x³ + 3x² - 10x -24`
When `x = -2`
`= x³ + 3x² - 10x -24`
`= (-2)³ + 3(-2)²- 10(-2) -24`
`= -8 + 12 + 20 - 24`
`= -32 + 32`
`= 0`
So by factor theorem, `(x + 2)` is a factor
On dividing` p(x)` by `(x + 2)`, we get` x² + x - 12`
Now, Using middle term splitting :
`= x² + x - 12`
`= x² + 4x - 3x - 12`
`= x(x+4) - 3(x+4)`
`= (x-3) (x+4)`