Problem 61 - Introductory Algebra for College Students

Question

Solve each equation and check your solutions. $$y^{3}+3 y^{2}+2 y=0$$

Step-by-Step Solution

Answer

The solutions of the equation are \( y = 0, -1, -2 \)

Step 1: Factor the equation

First of all, it's noticed that all terms in the equation share a common factor \( y \). Factoring this out gives: \( y(y^2 + 3y + 2) = 0 \).

Step 2: Find the factorization of the quadratic equation

Next, factorize the quadratic equation. \( y(y + 1)(y + 2) = 0 \)

Step 3: Solve for y

Now set each factor equal to zero and solve for \( y \). This leads to \( y=0, -1, -2 \)

Step 4: Check solutions

Finally, substitute each solution back into the original equation to ensure it holds true. You should get \(0\) for each solution.

Key Concepts

Factoring

Factoring is a method used to break down algebraic expressions into simpler parts, making them easier to solve or manipulate. In the context of solving equations, factoring is often used as an initial step to simplify expressions or to uncover roots of the equation.

• Identify common factors: The first step usually involves finding any common factors in the algebraic expression. For example, in the cubic equation \( y^3 + 3y^2 + 2y = 0 \), each term contains the variable \( y \). Thus, \( y \) can be factored out, resulting in \( y(y^2 + 3y + 2) = 0 \).
• Simplify further: After factoring out a common variable, check if the remaining expression can be further factored. In our case, the quadratic \( y^2 + 3y + 2 \) can still be factored into \( (y + 1)(y + 2) \).

Breaking down an equation through factoring not only simplifies it but also reveals the values of \( y \) that satisfy the equation when each factor is set to zero.

Cubic Equations

Cubic equations are polynomial equations where the highest power of the variable is three. The general form is \( ax^3 + bx^2 + cx + d = 0 \). These equations can sometimes be challenging to solve due to their higher degree. However, factoring simplifies the process significantly.

• Check for common factors: Identifying a common factor, as seen in \( y^3 + 3y^2 + 2y = 0 \), simplifies the equation.
• Reduce the degree: After factoring out common terms, focus on factoring the remaining polynomial, which usually reduces to a simpler quadratic equation. In this case, we simplified the cubic equation to a quadratic \( y^2 + 3y + 2 \).

Cubic equations often require an understanding of different factoring techniques, including grouping or synthetic division, and sometimes even numerical methods, to find solutions when factored forms are not obvious.

Solving Equations

Solving equations is a fundamental aspect of algebra that involves finding values for variables that satisfy the given equation. The process typically includes a sequence of transformation steps that isolate the variable on one side of the equation.

• Set factors to zero: Once the equation is factored, set each factor equal to zero. This is a result of the zero-product property, which states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).
• Solve simple equations: After setting each factor to zero, solve the resulting simple equations. For instance, from \( y(y + 1)(y + 2) = 0 \), we solve \( y = 0 \), \( y + 1 = 0 \), and \( y + 2 = 0 \), leading to solutions \( y = 0, -1, -2 \).
• Verify solutions: Always substitute the solutions back into the original equation to verify they are correct. Each calculated solution should satisfy the original equation.

By applying these techniques to solve equations, students can confidently find and verify solutions to a wide range of problems.