Question

use rational root theorum and the factor theorem to help solve the following equation
x^3+x^2-25x-25

Answer 1

Use synthetic to test the possible rational roots. You will find that -1 works and so x+1 is a factor and x^2+25 is the other factor.

Answer 2

Using the rational root theorem we can guess the roots of an equation
The rational roots maybe the factors of constant term over the coefficient of the highest power of x
so here the roots maybe
\[\frac{\pm 5 , 5,1}{1}\]
Let's check if +5 is a root
\[x^3+x^2-25x-25\]
x=5
\[5^3+5^2-25 \times 5-25=125+25-125-25=0\]
so 5 is a root or x-5 is a factor
We have now
\[x^3+x^2-25x-25\]
\[x^2(x-5)+6x(x-5)+5(x-5)\]so we have
\[(x^2+6x+5)(x-5)\]
In this quadratic x^2+6x+5
the roots maybe given as
\[\frac{\pm 5,1}{1}\]
Let's check if -5 is a root
\[x^2+6x+5=25-30+5=0\]
so -5 is a root or x+5 is a factor
Let's check -1 is a root, x=-1
\[1-6+5=0\]
so (x+1) is also a root
the roots are -1, 5 and -5