Solve each polynomial equation in Exercises 86–87. 2x^4 = 50 x^2

Verified step by step guidance

1. Rewrite the equation to set it equal to zero: Subtract \(50x^2\) from both sides to get \(2x^4 - 50x^2 = 0\).
2. Factor out the greatest common factor (GCF): Identify \(2x^2\) as the GCF and factor it out, resulting in \(2x^2(x^2 - 25) = 0\).
3. Apply the Zero Product Property: Set each factor equal to zero. This gives \(2x^2 = 0\) and \(x^2 - 25 = 0\).
4. Solve each equation separately: For \(2x^2 = 0\), divide both sides by 2 to find \(x^2 = 0\). For \(x^2 - 25 = 0\), add 25 to both sides to get \(x^2 = 25\).
5. Find the values of \(x\): Take the square root of both sides for each equation. For \(x^2 = 0\), \(x = 0\). For \(x^2 = 25\), \(x = \pm 5\).

Key Concepts

Polynomial Equations

A polynomial equation is an equation that involves a polynomial expression, which is a sum of terms consisting of variables raised to non-negative integer powers and coefficients. In this case, the equation 2x^4 = 50x^2 is a polynomial equation of degree 4, as the highest power of the variable x is 4. Understanding how to manipulate and solve polynomial equations is essential for finding the values of x that satisfy the equation.

Factoring

Factoring is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the context of the given equation, one can rearrange it to form a standard polynomial equation and then factor it to find the roots. This technique is crucial for solving polynomial equations efficiently, especially when they can be expressed as products of simpler polynomials.

Roots of Polynomial Equations

The roots of a polynomial equation are the values of the variable that make the equation true, typically where the polynomial equals zero. For the equation 2x^4 - 50x^2 = 0, finding the roots involves setting the equation to zero and solving for x. Understanding how to find and interpret these roots is fundamental in algebra, as they provide critical insights into the behavior of the polynomial function.