I am a bit stuck on factoring x^3 + 8x^2 + 17x + 10.

Question

I know my factor is (x + 2) so -2 is one of my roots. I then have to solve x^2 + 6x + 5 with the quadratic formula. After working out the problem, I got - 6 + or - radical -4/ 4. I'm confued on how to continue. I know that radical -4 is 2i.

Answer 1

The polynomial function f (x ) = x ³ + 8x ²+ 17x + 10

= x ³ + 6x ² + 2x ² + 12x  + 5x  + 10

= (x ³+ 6x ²+ 5x ) + (2x ² + 12x  + 10)

Factor out x  from the first group and factor out 2 from the second group.

= x (x ² + 6x  + 5) + 2(x ² + 6x  + 5)

x ³ + 8x ² + 17x  + 10 = (x ² + 6x  + 5)(x  + 2)

From factor theorem,

When x - c  is a factor of the polynomial then f (c ) = 0.

(x ² + 6x  + 5)(x  + 2) = 0

x  + 2 = 0 and x ² + 6x  + 5 = 0

x  = - 2 is a root.

Now solve the equation x ² + 6x + 5 = 0 using Quadratic formula.

Compare it to

Roots are

Roots are x  = -2, -1 and - 5.

Factoring of x ³ + 8x ²+ 17x + 10 = (x + 2)(x + 1)(x + 5).