Synthetic Division and the Factor Theorem

Question

Use the factor theorem to determine if the binomials given are factors of f(x). Use the binomials that are factors to write a factored form of f(x).

f(x)=x^3-7x-6;(x+2),(x-1)

Answer 1

If x+2 is a factor then x=-2 is a solution

-2 | 1 0 -7 -6

-2 4 6

___________________________

1 -2 -3 0

f(-2) = (-2)^3 - 7(-2) - 6

= -8 + 14 - 6

= 0

So -2 is a solution because synthetic division

shows the remainder is zero when f(x) is divided

by x+2, and the remainder theorem backs that up

since f(-2)=0. So Yes, DEFINITELY, UNDOUBTEDLY,

and CLEARLY, x=-2 is a solution and x+2 is a factor.

f(x) = (x+2)(x^2-2x-3)

= (x+2)(x-3)(x +1) <== factors!

So apprently x-1 is NOT a factor...

If it were, then x=1 WOULD be a solution and

synthetic division shows

1 | 1 0 -7 -6

1 1 -6

___________________________

1 1 -6 -12

that the remainder is -12 as does the

remainder theorem, since f(1) = 1^3 - 7(1) - 6

= 1 - 7 - 6

= -6-6

= -12

Finally, per the RATIONAL ROOT THEOREM,

P = +or- {1,2,3,6}

Q = +or- {1}

so P/Q = +or- {1,2,3,6}=P

and x=3 and x=-1 {the other two solutions,

are in the list

3 | 1 0 -7 -6

3 9 6

__________________________

1 3 2 0

and

-1 | 1 0 -7 -6

-1 1 6

_________________________

1 -1 -6 0

f(3) = 27 -21-6 = 0

f(-1) = -1 +7-6 = 0

So the complete solution set is x = {-2,3,-1}

and f(x) = (x+2)(x-3)(x+1)