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Solving the Equation (8x-4x^2-1)(x^2+2x+1)=4(x^2+x+1)
This equation involves a product of polynomials on the left-hand side and a simpler polynomial on the right-hand side. To solve it, we need to expand, simplify, and then find the roots of the resulting polynomial equation.
Step 1: Expanding and Simplifying
Let's first expand the left-hand side of the equation:
- (8x-4x^2-1)(x^2+2x+1) = -4x^4 + 4x^3 + 15x^2 + 16x + 1
Now, let's move all terms to one side of the equation:
- -4x^4 + 4x^3 + 15x^2 + 16x + 1 - 4(x^2+x+1) = 0
Simplifying further:
- -4x^4 + 4x^3 + 11x^2 + 12x - 3 = 0
Step 2: Finding the Roots
We now have a fourth-degree polynomial equation. Finding the roots of this equation can be challenging. Here are some common approaches:
- Factoring: Attempt to factor the polynomial. This might require some trial and error or using advanced factorization techniques.
- Rational Root Theorem: This theorem can help identify potential rational roots.
- Numerical Methods: Techniques like the Newton-Raphson method can be used to find approximate solutions.
In this case, factoring the polynomial might be the most straightforward approach. However, it's not immediately obvious how to factor this particular expression.
Conclusion
Solving the equation (8x-4x^2-1)(x^2+2x+1)=4(x^2+x+1) involves simplifying the equation into a fourth-degree polynomial equation. While finding the roots of a fourth-degree polynomial can be challenging, we can explore different approaches like factoring, using the rational root theorem, or employing numerical methods.