(x-5)(x-7)(x+4)(x+6)=504

4 min read Jun 17, 2024

Solving the Equation (x-5)(x-7)(x+4)(x+6) = 504

This equation presents a fun challenge involving factoring and solving a polynomial equation. Let's break it down step-by-step:

First, let's move the constant term to the left side of the equation: (x-5)(x-7)(x+4)(x+6) - 504 = 0
Now, let's try to simplify the expression by expanding the products: [(x-5)(x-7)][(x+4)(x+6)] - 504 = 0 (x² - 12x + 35)(x² + 10x + 24) - 504 = 0

Expand the product of the two quadratic expressions: x⁴ + 10x³ + 24x² - 12x³ - 120x² - 288x + 35x² + 350x + 840 - 504 = 0
Combine like terms: x⁴ - 2x³ - 61x² + 62x + 336 = 0

At this point, we need to factor the polynomial. This can be a bit tricky. One way is to try to find the roots of the polynomial. We can use the Rational Root Theorem to help us find potential rational roots. The Rational Root Theorem states that any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (336) and q is a factor of the leading coefficient (1).
By applying the Rational Root Theorem and testing different possibilities, we find that x = -4 is a root of the polynomial. This means that (x + 4) is a factor of the polynomial. We can then use polynomial long division or synthetic division to divide the polynomial by (x + 4).
Dividing the polynomial by (x + 4), we get: x⁴ - 2x³ - 61x² + 62x + 336 = (x + 4)(x³ - 6x² - 25x + 84)
Now, we need to factor the cubic polynomial (x³ - 6x² - 25x + 84). This can be done by grouping: x³ - 6x² - 25x + 84 = x²(x - 6) - 25(x - 6) = (x - 6)(x² - 25)
Finally, we can factor (x² - 25) as a difference of squares: (x² - 25) = (x - 5)(x + 5)

Combining all the factors, we have: (x + 4)(x - 6)(x - 5)(x + 5) = 0
Setting each factor equal to zero, we get the solutions: x = -4, x = 6, x = 5, x = -5

Therefore, the solutions to the equation (x-5)(x-7)(x+4)(x+6) = 504 are x = -4, x = 6, x = 5, and x = -5.