Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. $4 x^{3} - 12 x^{2} = 9 x - 27$

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First, rewrite the equation so that all terms are on one side, setting the equation equal to zero: $4x^3 - 12x^2 - 9x + 27 = 0$.

Next, group the terms to factor by grouping: group the first two terms and the last two terms separately, like this: $(4x^3 - 12x^2) - (9x - 27) = 0$.

Factor out the greatest common factor (GCF) from each group: from the first group factor out \$4x^2$, and from the second group factor out $9$, resulting in $4x^2(x - 3) - 9(x - 3) = 0$.

Since both terms contain the binomial $(x - 3)$, factor it out: $(x - 3)(4x^2 - 9) = 0$.

Now, apply the zero-product principle by setting each factor equal to zero: $x - 3 = 0$ and $4x^2 - 9 = 0$. Then solve each equation separately to find the values of $x$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding how to manipulate and simplify these expressions is essential for solving equations like 4x^3 - 12x^2 = 9x - 27.

Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials or factors. This step is crucial because it transforms the equation into a form where the zero-product principle can be applied to find the roots.

Zero-Product Principle

The zero-product principle states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for the variable to find the solutions of the equation.

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