My question is to solve the following polynomial equation by factoring: x^3+125=0

Question

That's all I got to be honest. 

Answer 1

125 = 5³ so what you have thereis the SUM OF CUBES. That is factorable; the first (linear, binomial) factor is x + 5.

The second factor (quadratic) is x² - 5x + 25. The general formula for factoring the sum of cubes is in your textbook and really must be MEMORIZED.

 

Now, look also at the formula for factoring the difference of cubes.

 

Note: the first factor's sign (interior) is same as that of the original binomial of degree three, and the quadratic factor is such that the sign of the middle of its three terms is the opposite of that interior sign.

 

So that, whether you are factoring the sum of cubes or the difference of cubes, altogether there will be exactly one minus sign in that factorization.

Answer 2

If we don't know the special cubic factoring methods, we can use the root x=-5 and use synthetic division to find the other factor of the polynomial.

 

-5 | 1       0        0         125

 

              -5       25       -125

__________________________

      1      -5       25           0

 

 

We get a factor polynomial equation of

 

(x + 5)(x² - 5x + 25) = 0

 

 

Now it becomes easier to solve.

Answer 3

x³ + 125 = 0

x³ + 5³ = 0

 

Look up "sum of cubes" factoring.