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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Key Concepts

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

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or expressions. For the given equation, common factors like powers of x and constants can be factored out to simplify the equation, making it easier to solve.

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.

Solving Polynomial Equations

Solving polynomial equations involves finding all values of the variable that satisfy the equation. After factoring and applying the zero-product principle, each resulting equation is solved individually to find all possible roots.

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