Answer:
[tex]x^3 - 10 x^2 -53x -42 = ( x + 1)(x + 3) (x - 14)[/tex]
Step-by-step explanation:
[tex]x^3 - 10x^2 -53x - 42[/tex]
The factors of 42 : ±1 , ±2 , ±3 , ±6, ±7, ±14, ±21 , ±42
Using trial and error method we will find the first root of the polynomial.
1 : ( 1 )³ - 10 ( 1 )² - 53 ( 1 ) - 42
1 - 10 - 53 - 42 ≠ 0
Therefore 1 is not a root
-1 : ( -1 )³ - 10 (- 1 )² - 53 (- 1 ) - 42
- 1 - 10 + 53 - 42
53 - 53 = 0
Therefore - 1 is a root of the polynomial.
Therefore ( x + 1) is a factor.
Now by long division or by using synthetic division we can find other factors.
Synthetic Division :
-1 | 1 -10 -53 - 42
| 0 - 1 11 42
|______________________
1 - 11 - 42 0
Therefore ,
[tex]x^3 - 10 x^2 -53x -42 = ( x + 1)(x^2 - 11x +42)[/tex] ------ ( 1 )
Next further factorize x² - 11x - 42
x² - 11x - 42
= x² - 14x + 3x - 42
=x(x - 14) + 3 (x - 14)
=(x + 3 )(x - 14)
Therefore ,
[tex]x^3 - 10 x^2 -53x -42 = ( x + 1)(x + 3) (x - 14)[/tex]
