Problem 32 - Precalculus: Functions and Graphs

Question

Solve the equation. \(y^{4 / 3}=-3 y\)

Step-by-Step Solution

Answer

The solutions are \( y = 0 \) and \( y = -3 \).

Step 1: Set the Equation to Zero

Rearrange the equation so that everything is on one side. We have: \[ y^{4/3} + 3y = 0 \]

Step 2: Factor Out the Common Term

Identify the common factor from the terms on the left side of the equation. Factoring out \(y^{1/3}\), we get: \[ y^{1/3} (y + 3) = 0 \]

Step 3: Apply the Zero Product Property

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero and solve: 1. \( y^{1/3} = 0 \)2. \( y + 3 = 0 \)

Step 4: Solve Each Factor

Solve each equation from Step 3: 1. For \( y^{1/3} = 0 \), cube both sides to get \( y = 0 \).2. For \( y + 3 = 0 \), subtract 3 from both sides to get \( y = -3 \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a technique used to simplify equations, making them easier to solve. When you have an equation, the goal is often to factor out common terms. This means you need to identify terms that appear in all parts of the polynomial expression.

• To factor, look for the greatest common factor (GCF) between terms. In our exercise, the GCF is determined by the lowest power of any common variable, or by constants if applicable.
• For the equation \(y^{4/3} + 3y = 0\), notice how each term includes a 'y' factor. The smallest power here is \(y^{1/3}\).
• By factoring out \(y^{1/3}\), we separate the equation into simpler components: \(y^{1/3}(y + 3) = 0\).

Factoring reduces complex expressions to products of simpler expressions, which is critical in solving algebraic equations efficiently. This method is especially useful before applying properties like the zero product property.

Zero Product Property

The zero product property is a fundamental concept in algebra that makes solving polynomial equations straightforward once they have been factored. It states that if the product of two or more terms equals zero, at least one of the terms must be zero.

• This concept is highly useful, particularly with expressions like \(a \times b = 0\).
• To apply this property, take the factors of an equation and individually set each equal to zero.
• For the factored equation \(y^{1/3}(y + 3) = 0\), apply the zero product property to solve the factors separately: \(y^{1/3} = 0\) and \(y + 3 = 0\).

This approach allows us to find each possible solution to the original polynomial equation. It simplifies the process by breaking the problem into smaller, more manageable segments.

Cubic Equations

Cubic equations are those where the highest degree of its variable is three. An understanding of cubic equations includes knowing how to solve for their roots using various algebraic methods.

• Our original equation began with a fractional exponent: \(y^{4/3}\).
• By rewriting it as \(y^{1/3}(y + 3) = 0\), you solve a cubic form by considering \(y^{1/3}\) as a variable to the power of one.
• In terms of solutions, cubic equations can have one real root or three real roots depending on the specific format and coefficients involved.

In solving cubic equations, such as through factoring, you transform them into simpler, linear equations or quadratic forms that allow for easy solution, providing straightforward methods to find their roots. Cubic equations often arise in various mathematical contexts and mastering their solution paves the way for solving more complex equations.