Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. $4y^3-2=y-8y^2$

Verified step by step guidance

1. First, rewrite the equation so that all terms are on one side, setting the equation equal to zero: \$4y^3 - 2 - y + 8y^2 = 0$.
2. Combine like terms to simplify the expression: \$4y^3 + 8y^2 - y - 2 = 0$.
3. Group terms to factor by grouping: group the first two terms and the last two terms separately: \((4y^3 + 8y^2) + (-y - 2) = 0\).
4. Factor out the greatest common factor (GCF) from each group: \$4y^2(y + 2) - 1(y + 2) = 0$.
5. Since both groups contain the factor \((y + 2)\), factor it out: \((y + 2)(4y^2 - 1) = 0\). Then, apply the zero-product principle by setting each factor equal to zero: \(y + 2 = 0\) and \$4y^2 - 1 = 0$.

Key Concepts

Polynomial Equations

Polynomial equations are algebraic expressions set equal to zero, involving variables raised to whole-number exponents. Understanding how to manipulate and simplify these expressions is essential for solving them. In this problem, the equation involves a cubic polynomial, which requires careful rearrangement before factoring.

Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials or factors. This step is crucial because it breaks down complex expressions into manageable parts. Common factoring techniques include factoring out the greatest common factor, grouping, and special products like difference of squares.

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, providing the solutions to the polynomial equation after factoring.