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To factor the polynomial x^3 - 343, we recognize that 343 is the cube of 7. Therefore, we can rewrite the polynomial as (x)^3 - (7)^3.
Using the formula for factoring the difference of cubes, we have:
(x - 7)(x^2 + 7x + 49)
So, the factored form of the polynomial is (x - 7)(x^2 + 7x + 49).
Next, to find the real number solutions of the equation x^3 = 343, we can set the equation equal to zero:
x^3 - 343 = 0
We can rearrange this equation as:
x^3 = 343
Taking the cube root of both sides, we get:
x = β343
Simplifying, we find that x = 7.
Therefore, the real number solution of the equation x^3 = 343 is x = 7.