Question

Solve : x^3+3x^2-25x-75=0

Answer 1

`x=3, x=\pm5`

Explanation:

Let's start with compiling the list: all possible rational roots are in the list composed dividing the divisor of the lead term (75 in this case) by the divisors of the lead term, in this case 1 - not counting their sign. In our case divisors of 75 are 1, 3, 5, 15, 25 and 75 itself. Divisors of 1 are, obviously, just 1. The list is thus
`\pm1, \pm3, \pm5, \pm15, \pm25, \pm75`. Now it's a matter of trial and error. In particular, for
`x=5` we get

`5^3+3 (5^2)-25(5)-75= 125+75-125-75 = 0`

Thus, we can divide by a factor of
`(x-5)` - use the method you prefer, I find faster to just rewrite everything as
`(x-5)(Ax^2+Bx+C)`, multiply it out and find the values of the coefficients. Either method you use, you should be able to rewrite the polynomial as

`(x-5)(x^2+8x+15)=0`

Now you have two choice. Either apply the quadratic formula or find two numbers that add to 8 and multiply to 15 - namely, 3 and 5. You can rewrite the second factor as

`(x+3)(x+5)`, which makes it faster to solve.

Finally, the three solutions are 3, 5 and -5

Answer 2

Answer: x = -3, -5, 5 hope thats correct

Explanation: