(x%2B5)(x%2B8)(x%2B2)-880%3D0.jpeg "(x-1)(x+5)(x+8)(x+2)-880=0")
Solving the Quartic Equation: (x-1)(x+5)(x+8)(x+2) - 880 = 0
This article will guide you through the process of solving the quartic equation:
(x-1)(x+5)(x+8)(x+2) - 880 = 0
Understanding the Equation
This equation is a quartic equation, meaning it has a highest power of 4. It's presented in a factored form, which can be helpful in finding solutions. However, the presence of the constant term (-880) makes it more challenging.
Solving the Equation
-
Expand the product: Begin by expanding the product of the factors on the left side:
(x-1)(x+5)(x+8)(x+2) = (x² + 4x - 5)(x² + 10x + 16)
Now, expand further:
x⁴ + 14x³ + 49x² + 48x - 80 = 0
-
Look for possible rational roots: We can use the Rational Root Theorem to identify potential rational roots. The 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 (-80) and q is a factor of the leading coefficient (1).
Factors of -80: ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80 Factors of 1: ±1
Therefore, the possible rational roots are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±80.
-
Test the possible roots: We can use synthetic division or direct substitution to test these possible roots. We find that x = 2 is a root:
(2)⁴ + 14(2)³ + 49(2)² + 48(2) - 80 = 16 + 112 + 196 + 96 - 80 = 240 - 80 = 160 ≠ 0
We need to continue testing other roots. By testing, we find that x = -5 is also a root.
-
Factor the equation: Since we found two roots, we can factor the equation:
(x - 2)(x + 5)(x² + 6x + 8) = 0
The quadratic factor (x² + 6x + 8) can be factored further:
(x - 2)(x + 5)(x + 2)(x + 4) = 0
-
Solve for the remaining roots: Now, we can easily solve for the remaining roots:
x - 2 = 0 => x = 2 x + 5 = 0 => x = -5 x + 2 = 0 => x = -2 x + 4 = 0 => x = -4
Conclusion
Therefore, the solutions to the quartic equation (x-1)(x+5)(x+8)(x+2) - 880 = 0 are:
x = 2, x = -5, x = -2, and x = -4