Questions: Solve the polynomial equation by factoring and then using the zero-product principle. 27x-9=3x^3-x^2 Find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Use a comma to separate answers as needed. Type an integer or a simplified fraction.) B. There is no solution.

$Solve the polynomial equation by factoring and then using the zero-product principle. 27x-9=3x^3-x^2 Find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Use a comma to separate answers as needed. Type an integer or a simplified fraction.) B. There is no solution.$

Transcript text: Solve the polynomial equation by factoring and then using the zero-product principle. \[ 27 x-9=3 x^{3}-x^{2} \] Find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your ch A. The solution set is $\square$ \} (Use a comma to separate answers as needed. Type an integer or a simplified fraction.) B. There is no solution.

Solution

Solution Steps

To solve the polynomial equation $27x - 9 = 3x^3 - x^2$ by factoring and then using the zero-product principle, follow these steps:

Move all terms to one side of the equation to set it to zero.
Factor the resulting polynomial.
Use the zero-product principle to find the roots of the equation.

Step 1: Move All Terms to One Side

First, we move all terms to one side of the equation to set it to zero: \[ 27x - 9 = 3x^3 - x^2 \] \[ 27x - 9 - 3x^3 + x^2 = 0 \] \[ -3x^3 + x^2 + 27x - 9 = 0 \]

Step 2: Factor the Polynomial

Next, we factor the polynomial $-3x^3 + x^2 + 27x - 9$. The factored form is: \[ -3(x + 3)(x - \frac{1}{3})(x - 3) = 0 \]

Step 3: Apply the Zero-Product Principle

Using the zero-product principle, we set each factor equal to zero and solve for $x$: \[ x + 3 = 0 \implies x = -3 \] \[ x - \frac{1}{3} = 0 \implies x = \frac{1}{3} \] \[ x - 3 = 0 \implies x = 3 \]