(x+2)(x-5)(x-6)(x+1)=144

5 min read Jun 16, 2024

Solving the Equation (x+2)(x-5)(x-6)(x+1) = 144

This equation presents a challenge because it's a fourth-degree polynomial equation. Let's break down the steps to solve it:

1. Expand the Equation

First, expand the left side of the equation by multiplying the factors together.

Step 1: Multiply the first two factors: (x+2)(x-5) = x² - 3x - 10
Step 2: Multiply the last two factors: (x-6)(x+1) = x² - 5x - 6
Step 3: Multiply the results from steps 1 and 2: (x² - 3x - 10)(x² - 5x - 6) = x⁴ - 8x³ - 7x² + 90x + 60

Now our equation becomes: x⁴ - 8x³ - 7x² + 90x + 60 = 144

2. Rearrange into Standard Form

Subtract 144 from both sides to get a standard form polynomial equation:

x⁴ - 8x³ - 7x² + 90x - 84 = 0

3. Finding Solutions

Now we have a fourth-degree polynomial equation. Unfortunately, there isn't a simple formula like the quadratic formula for solving equations of this degree. Here are the common approaches:

Factoring: Try to factor the polynomial. This might be possible if you can find patterns or use techniques like grouping. However, it's not always straightforward with higher-degree polynomials.
Rational Root Theorem: This theorem helps identify potential rational roots (roots that can be expressed as fractions). You can use this theorem to test possible roots and try to factor the polynomial further.
Numerical Methods: For more complex equations, numerical methods like the Newton-Raphson method can approximate the solutions. These methods involve iterative processes to get closer and closer to the actual roots.
Graphing: Plotting the graph of the function can help visually identify the approximate locations of the roots (where the graph intersects the x-axis).

4. Solution Possibilities

It's important to remember that a fourth-degree polynomial can have up to four solutions. These solutions could be real numbers or complex numbers.

Example: Solving Using the Rational Root Theorem

Let's try the Rational Root Theorem to get started.

The Rational Root Theorem states that any rational root of a polynomial with integer coefficients must be of the form p/q, where p is a factor of the constant term (in this case, -84) and q is a factor of the leading coefficient (in this case, 1).

Factors of -84: ±1, ±2, ±3, ±4, ±6, ±7, ±12, ±14, ±21, ±28, ±42, ±84
Factors of 1: ±1

Therefore, potential rational roots are all possible combinations of these factors.

We can test these possible roots using synthetic division or by substituting them into the equation.

For example, testing x = 2: 2⁴ - 8(2)³ - 7(2)² + 90(2) - 84 = 0

Since this results in 0, we know that x = 2 is a root. This means (x - 2) is a factor of the polynomial. We can now use polynomial division or other methods to factor the equation further and find the remaining roots.

Conclusion

Solving the equation (x+2)(x-5)(x-6)(x+1) = 144 involves several steps, including expanding the equation, rearranging into standard form, and then using various techniques to find the roots. Remember that there might be multiple solutions, and depending on the equation's complexity, finding them might require numerical methods or advanced factorization techniques.