(x-2)(x^2+2x+1)=4(x+1)

3 min read Jun 17, 2024

Solving the Equation: (x-2)(x^2+2x+1) = 4(x+1)

This equation involves a combination of multiplication and factoring, making it an excellent example of a problem that requires strategic manipulation to solve. Here's how to approach it:

1. Expand the Left Side

Start by expanding the left side of the equation using the distributive property:

(x-2)(x^2+2x+1) = x(x^2+2x+1) - 2(x^2+2x+1)

Expanding further, we get:

x^3 + 2x^2 + x - 2x^2 - 4x - 2 = 4x + 4

2. Simplify the Equation

Combine like terms on both sides:

x^3 - 3x - 2 = 4x + 4

Move all terms to one side to set the equation equal to zero:

x^3 - 7x - 6 = 0

3. Factor the Cubic Equation

Unfortunately, there's no easy formula for factoring cubic equations. We need to employ some trial and error and use the Rational Root Theorem to find potential roots.

The Rational Root Theorem suggests that any rational root of the equation must be a divisor of the constant term (-6) divided by a divisor of the leading coefficient (1). This gives us potential roots: ±1, ±2, ±3, ±6.

By testing these potential roots, we find that x = -1 is a root of the equation. This means (x+1) is a factor of the cubic polynomial.

We can use polynomial long division or synthetic division to divide the cubic polynomial by (x+1). This results in:

(x + 1)(x^2 - x - 6) = 0

Now we can factor the quadratic expression:

(x + 1)(x - 3)(x + 2) = 0

4. Solve for x

Finally, we can solve for the roots by setting each factor equal to zero:

x + 1 = 0 => x = -1 x - 3 = 0 => x = 3 x + 2 = 0 => x = -2

Therefore, the solutions to the equation (x-2)(x^2+2x+1) = 4(x+1) are x = -1, x = 3, and x = -2.