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)=2x^3+x^2-50x-25;(x+5),(x-5)

Answer 1

if x+5 is a factor then x=-5 is a solution

-5 | 2 1 -50 -25

-10 45 25

__________________________

2 -9 -5 0

Yes, -5 is a solution because the remainder is ZER0;

note that f(-5) = 2(-5)^3 +(-5)^2 - 50(-5) - 25

= 2(-125) + 25 + 250 -25

= -250 + 25 + 250 - 25

= 0

So f(x) = (x+5)(2x^2 - 9x - 5)

If x-5 is a factor then x=5 is a solution..

5 | 2 -9 -5

10 5

________________________

2 1 0

So yes, 5 is a solution because the remainder is ZER0

AND f(5) = 2(5^3) + 5^2 - 50(5) - 25

= 2(125) + 25 - 250 -25

= 250 + 25 - 250 -25

= 0

So f(x) = (x+5)(x-5)(2x+1)

Finally note that, per the Rational Root Theorem,

P = +or-{1,5,25} and Q = +or-{1,2}

so the set of POSSIBLE/POTENTIAL rational solutions

is +or- { 1, 1/2 , 5 , 5/2, 25, 25/2}

AND since 2x+1 is the last factor, x = -1/2 is

the corresponding solution which IS IN THE LIST!!

Finally

f(-1/2) = 2(-1/2)^3 + (-1/2)^2 - 50(-1/2) - 25

= 2(-1/8) + (1/4) + 25 - 25

= -1/4 + 1/4 + 25-25

= 0

So f(x) = (x+5)(x-5)(2x+1)

and the solution set is x={5,-5,-1/2}