Now, we need to find the roots of this polynomial equation. We can use numerical methods or factorize the equation using rational roots theorem. Let's factorize this equation using rational roots theorem:
The possible rational roots are factors of the constant term, 120, divided by factors of the leading coefficient, 1. ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±24, ±30, ±40, ±60, ±120
By trying out these values, we find that x = 5 is a root of the equation:
Therefore, x = 5 is a root of the equation. To find the other roots, we can divide the polynomial by (x - 5) using long division or synthetic division to find the quadratic equation that makes the next factor.
To solve this equation, we need to expand the left side of the equation by multiplying the factors:
(x-2)(x-3)(x+4)(x+5) = 1320
(x^2 - 3x - 2x + 6)(x^2 + 5x + 4x + 20) = 1320
(x^2 - 5x + 6)(x^2 + 9x + 20) = 1320
x^4 + 9x^3 + 20x^2 - 5x^3 - 45x^2 - 100x + 6x^2 + 54x + 120 = 1320
x^4 + 4x^3 - 19x^2 + 74x + 120 = 1320
x^4 + 4x^3 - 19x^2 + 74x - 120 = 0
Now, we need to find the roots of this polynomial equation. We can use numerical methods or factorize the equation using rational roots theorem. Let's factorize this equation using rational roots theorem:
The possible rational roots are factors of the constant term, 120, divided by factors of the leading coefficient, 1.
±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±24, ±30, ±40, ±60, ±120
By trying out these values, we find that x = 5 is a root of the equation:
(5)^4 + 4(5)^3 - 19(5)^2 + 74(5) - 120 = 0
625 + 500 - 475 + 370 - 120 = 0
125 + 70 + 370 - 120 = 0
195 + 370 - 120 = 0
565 - 120 = 0
445 = 0
Therefore, x = 5 is a root of the equation. To find the other roots, we can divide the polynomial by (x - 5) using long division or synthetic division to find the quadratic equation that makes the next factor.