find all real solutions to the equation.\n\n$x^{3}-5x^{2}-7x + 35 = 0$

Explanation:

Step1: Group the terms

Group the given equation (x^{3}-5x^{2}-7x + 35=0) as ((x^{3}-5x^{2})+(-7x + 35)=0). Factor out (x^{2}) from the first group and (-7) from the second group: (x^{2}(x - 5)-7(x - 5)=0).

Step2: Factor by grouping

Using the distributive property (a(b - c)-d(b - c)=(a - d)(b - c)), we get ((x^{2}-7)(x - 5)=0).

Step3: Solve each factor

Set (x-5 = 0), then (x=5). Set (x^{2}-7=0), then (x^{2}=7), and (x=\pm\sqrt{7}).

Answer:

(x = 5,x=\sqrt{7},x=-\sqrt{7})