Rational root theorem

Question

How do you use the rational root theorem to factor a cube?

x^3+4x^2-15x-18

-x^3-7x^2+14x+48

Answer 1

x^3 +4x^2 -15x -18 = 0

try integer factors of 18, start with 1 and -1

(-1)^3 +4(-1)^2 -15(-1)-18 = -1+4+15-18= 18-18=0

-1 is a zero. x+1 is a factor

divide the cubic by x+1 to get

x^2 +3x-18

look for integer factors of 18, 6 and 3

(x+6)(x-3)

the cubic polynomial factors to (x+1)(x+6)(x-3)

which is 2 negative zeros and 1 positive zero

what you may suspect from descartes rule on alternating signs

the cubic has 1 alternating sign, so expect maximum 1 positive real zero

replace x by -x and the cubic has 2 alternating signs, so expect max 2 negative real zeros

or use a graphing calculator which shows x intercepts at -6,-1 and 3