Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. $5 x^{4} - 20 x^{2} = 0$

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Start by writing the given polynomial equation: $5x^4 - 20x^2 = 0$.

Factor out the greatest common factor (GCF) from both terms. Identify the GCF of \$5x^4$ and \$20x^2$, which is \$5x^2$. So, factor it out: $5x^2(x^2 - 4) = 0$.

Recognize that the expression inside the parentheses, $x^2 - 4$, is a difference of squares. Factor it further as $(x - 2)(x + 2)$, so the equation becomes $5x^2(x - 2)(x + 2) = 0$.

Apply 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: $5x^2 = 0$, $x - 2 = 0$, and $x + 2 = 0$.

Solve each equation separately: For $5x^2 = 0$, divide both sides by 5 and solve for $x$. For $x - 2 = 0$, add 2 to both sides. For $x + 2 = 0$, subtract 2 from both sides. These solutions give the roots of the original polynomial.

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