Solve the polynomial equation.
15x^3-119x^2-10x+16=0
try x = ±1 , ±2, ±4, ±8
I got x = 8 to work for me
so after a sythetic division, I got
(x-8)(15x^2 + x - 2) = 0
(x-8)(3x-1)(5x + 2) = 0
x = 8 , 1/3 , -2/5
Okay, thank you.
To solve the given polynomial equation, 15x^3 - 119x^2 - 10x + 16 = 0, we can use the technique called factoring.
Step 1: Factor out the greatest common factor, if possible.
In this case, there is no common factor greater than 1 for all the terms. So, we can skip this step.
Step 2: Check for rational roots using the Rational Root Theorem.
According to the Rational Root Theorem, the potential rational roots are all the divisors of the constant term (16) divided by the divisors of the leading coefficient (15):
±1, ±2, ±4, ±8, ±16, ±1/3, ±2/3, ±4/3, ±8/3, ±16/3
Step 3: Use synthetic division or long division to find the roots.
By trying the potential rational roots, we can find that x = 2 is a root of the polynomial equation. Using synthetic division, we can divide the polynomial by (x - 2) to obtain the remaining quadratic equation:
(x - 2)(15x^2 - 89x - 8) = 0
Step 4: Solve the quadratic equation.
The remaining quadratic equation 15x^2 - 89x - 8 = 0 can be solved using factoring, completing the square, or the quadratic formula. In this case, we will use factoring:
(3x + 1)(5x - 8) = 0
Setting each factor equal to zero, we have two potential solutions:
3x + 1 = 0 -> x = -1/3
5x - 8 = 0 -> x = 8/5
Therefore, the roots of the polynomial equation 15x^3 - 119x^2 - 10x + 16 = 0 are x = 2, x = -1/3, and x = 8/5.