Question

Show that (x + 4), (x – 3) and (x – 7) are factors of x³ – 6x² – 19x + 84.

Answer 1

Let f(x) = x³ – 6x² – 19x + 84 

If x + 4 = 0, then x = -4 

If x – 3 = 0, then x = 3 

and if x – 7 = 0, then x = 7

Now, 

f(-4) = (-4)³ – 6(-4)² – 19(-4) + 84 

= -64 – 96 + 76 + 84 

= 160 – 160 

= 0 

f(-4) = 0

f(3) = (3)³ – 6(3)² – 19 x 3 + 84 

= 27 – 54 – 57 + 84 

= 111 -111

= 0 

f(3) = 0 

f(7) = (7)³ – 6(7)² – 19 x 7 + 84 

= 343 – 294 – 133 + 84

= 427 – 427 

= 0 

f(7) = 0 

Hence (x + 4), (x – 3), (x – 7) are the factors of f(x).