Question

List all possible (or potential) rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely over the real numbers. Please show all of your work.
f(x)=x^3+3x^2-15x+10
List all possible (or potential) rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely over the real numbers. Please show all of your work.
f(x)=x^3+3x^2-15x+10
@Mathematics

Answer 1

The possible rational zeros are the factors of the constant term, which is 10, divided by the factors of the coefficient of the leading term, which is 1. So +_10/1, +-5/1,+-2/1 or 10,-10,5,-5,2,-2

Answer 2

Also +1,-1

Answer 3

Use synthetic division to find out if any yield a remainder of 0: 2| 1 3 -15 10
2 10 -10
1 5 -5 0

So x-2 is a factor and the other factor is given by the bottom row of the synthetic division process and is x^2+5x_5

So far we know that x^3 + 3x - 15x + 10 = (x-2)(x^2+5x-5)
Now x^2 +5x -5 will not factor by the usual methods so we will use the quadratic formula to see if there are any more real roots: x =[ -5 +- sqrt(5^2 -4(1)(-5)]/2(1)
x = (-5+- sqrt 45)/2=(-5+-3sqrt5)/2
So yes, there are 2 more real roots and so x^3+3x-15x+10 = (x-2)[x-(-5+3sqrt5)/2][x-(-5-3sqrt5)/2]

Answer 4

Thank you Mertsj.

Answer 5

x^2+5x_5 <<<< did you mean -5?