Problem 28 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$2 x^{3}=50 x$$

Step-by-Step Solution

Answer

The solutions are \(x = 0, x = 5, x = -5\).

Step 1: Move all terms to one side of the equation

Start by rewriting the equation with all terms set to zero: \[ 2x^3 - 50x = 0 \] This allows us to factor the left-hand side.

Step 2: Factor out the greatest common factor

Look for the greatest common factor (GCF) in the equation. The GCF is \(2x\). Factor \(2x\) out from the left side of the equation:\[ 2x(x^2 - 25) = 0 \]

Step 3: Apply the difference of squares

Recognize that \(x^2 - 25\) is a difference of squares, which can be factored further using the formula \(a^2 - b^2 = (a - b)(a + b)\):\[ 2x(x - 5)(x + 5) = 0 \]

Step 4: Solve for each factor

Set each factor equal to zero to solve for \(x\):1. \(2x = 0 \) leads to \(x = 0\).2. \(x - 5 = 0 \) leads to \(x = 5\).3. \(x + 5 = 0 \) leads to \(x = -5\).

Step 5: Combine solutions

Combine all solutions from all factors. Therefore, the solutions to the equation \(2x^3=50x\) are \(x = 0, x = 5,\) and \(x = -5\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial step in the factoring process. It involves finding the highest factor that divides all terms of a polynomial. By factoring out the GCF, we simplify the polynomial, making further factoring easier.When looking at the equation \(2x^3 - 50x = 0\), the terms \(2x^3\) and \(-50x\) both share the factor \(2x\). By factoring \(2x\) out of both terms, we rewrite the equation as:

• \(2x(x^2 - 25) = 0\)

This simplification is often the first step in solving polynomial equations, as it reduces the complexity of the expression, allowing us to apply other techniques such as factoring special patterns.

Difference of Squares

The Difference of Squares is a pattern used in factoring specific quadratics. It applies when you have a polynomial in the form \(a^2 - b^2\). The factorization of this form is \((a - b)(a + b)\).In the previous step, after factoring out the GCF, the expression \(x^2 - 25\) presents itself as a difference of squares:

• \(x^2 - 25 = (x - 5)(x + 5)\)

Notice how both \(x^2\) and \(25\) are perfect squares, which enables us to apply the difference of squares formula. Recognizing this pattern simplifies the process further, as it breaks down the quadratic into linear factors. This makes solving the equation manageable by setting each factor to zero.

Solving Polynomial Equations

Solving polynomial equations involves setting each factor of the polynomial equal to zero and solving for the variable. Once we have factored the polynomial completely, as we did with \(2x(x - 5)(x + 5) = 0\), the next steps involve:- Setting each factor equal to zero:

• \(2x = 0\)
• \(x - 5 = 0\)
• \(x + 5 = 0\)

- Solving each resulting equation:

• For \(2x = 0\), divide by \(2\) to get \(x = 0\).
• For \(x - 5 = 0\), add \(5\) to both sides to find \(x = 5\).
• For \(x + 5 = 0\), subtract \(5\) to obtain \(x = -5\).

Once these solutions are found, combine them to get the complete solution set \(x = 0, x = 5, x = -5\). This method ensures all solutions to the equation are identified.