Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. $3 x^{4} = 81 x$

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Start by rewriting the equation to set it equal to zero: $3x^4 - 81x = 0$.

Factor out the greatest common factor (GCF) from the left side. Identify the GCF of \$3x^4$ and \$81x$, which is \$3x$, and factor it out: $3x(x^3 - 27) = 0$.

Recognize that $x^3 - 27$ is a difference of cubes, since $27 = 3^3$. Use the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, where $a = x$ and $b = 3$.

Apply the formula to factor $x^3 - 27$ as $(x - 3)(x^2 + 3x + 9)$, so the full factorization is $3x(x - 3)(x^2 + 3x + 9) = 0$.

Use the zero-product principle, which states that if a product of factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero: $3x = 0$, $x - 3 = 0$, and $x^2 + 3x + 9 = 0$, then solve each equation for $x$.

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