Problem 5 - College Algebra

Question

Solve each polynomial equation in by factoring and then using the zero-product principle. $$ 2 x-3=8 x^{3}-12 x^{2} $$

Step-by-Step Solution

Answer

A non-trivial factorization of the polynomial seems not to be achievable, hence the solutions of the equation, if they exist, are likely to be complex and would require a different method, such as using the cubic formula or numerical methods.

Step 1: Rearrange the equation

First, we rearrange the given equation to the standard polynomial form. That is, all terms should be on one side of the equality with zero on the other side. This gives us the equation \(8x^{3} - 12x^{2} - 2x + 3 = 0\).

Step 2: Factorize the equation

Next, we factorize the equation. As there is not a direct standard form for factorization, we can divide through by the leading coefficient 2, which makes it simpler for factorization. Here, the equation becomes \(4x^{3} - 6x^{2} - x + 3/2 = 0\).

Step 3: Apply the zero-product principle

Now, we apply the zero-product principle. The solutions (roots) of the polynomial are the values that make the polynomial equal to zero. Those are values of \(x\) that make \(4x^{3} - 6x^{2} - x + 3/2 = 0\). To find these solutions, the polynomial should be factored into forms like \((px - q)(rx - s) = 0\), and then equate each factor individually to zero to solve for \(x\). However, in this particular case, it seems that a non-trivial factorization may not be achievable.