Problem 53 - Elementary Algebra

Question

(a) solve. (b) check. $$ w^{3}+6 w^{2}+8 w=0 $$

Step-by-Step Solution

Answer

The solutions are \(w = 0, -2, -4\).

Step 1 - Factor Out the Greatest Common Factor (GCF)

Identify the greatest common factor (GCF) in the equation. In this equation, the GCF is \(w\). Factoring out \(w\) gives: \[ w(w^2 + 6w + 8) = 0 \]

Step 2 - Solve for the First Solution

Set the factored term \(w\) equal to zero and solve for \(w\): \[ w = 0 \] This is the first solution.

Step 3 - Factor the Quadratic Expression

Now consider the quadratic expression \(w^2 + 6w + 8\). Factor this expression by finding two numbers that multiply to 8 and add up to 6. These numbers are 2 and 4. So, we factor as follows: \[ w^2 + 6w + 8 = (w + 2)(w + 4) \]

Step 4 - Solve for the Remaining Solutions

Set each factor equal to zero and solve for \(w\): \(w + 2 = 0\): \[ w = -2 \] \(w + 4 = 0\): \[ w = -4 \] These are the remaining solutions.

Step 5 - Verify the Solutions

Substitute each solution back into the original equation: For \(w = 0\): \[ 0^3 + 6 \times 0^2 + 8 \times 0 = 0 \] For \(w = -2\): \[ (-2)^3 + 6 \times (-2)^2 + 8 \times (-2) = -8 + 24 - 16 = 0 \] For \(w = -4\): \[ (-4)^3 + 6 \times (-4)^2 + 8 \times (-4) = -64 + 96 - 32 = 0 \] All of the solutions satisfy the original equation.

Key Concepts

Greatest Common Factor (GCF)

In polynomial equations, finding the Greatest Common Factor (GCF) is a crucial first step in simplifying and solving the equation.
The GCF is the largest factor that can divide each term of the polynomial without leaving a remainder.
Here's how to identify and factor out the GCF in the given polynomial equation.
Let's look at the example: \( w^3 + 6w^2 + 8w = 0 \).

Step 1: Identify the GCF
- Here, each term has a common factor of \( w \).
Step 2: Factor out the GCF
- Factoring out \( w \) gives us: \[ w(w^2 + 6w + 8) = 0 \]

By factoring out the GCF, we have simplified our original polynomial into a product of simpler factors. This helps make solving for \( w \) easier!

Quadratic Expression

A quadratic expression is a polynomial of degree 2, generally represented as \( ax^2 + bx + c \).
The process of factoring a quadratic expression involves rewriting it as a product of two binomials.

In our example: \( w^2 + 6w + 8 \), we need to factor this quadratic polynomial.

Step 1: Look for two numbers that multiply to 8 and add up to 6.
- These numbers are 2 and 4.
Step 2: Rewrite the expression as a product of binomials.
- We can factor it as follows: \[ w^2 + 6w + 8 = (w + 2)(w + 4) \]

By factoring the quadratic polynomial, we decompose it into simpler, solvable parts. Now we can find the solutions for \( w \) by setting each binomial equal to zero.

Verifying Solutions

After finding the solutions of a polynomial equation, it is essential to verify that each solution satisfies the original equation.
This ensures that our solutions are correct and that no mistakes were made during the solving process.

Let's verify the solutions we found: \( w = 0, w = -2, w = -4 \).

Substitute each solution back into the original polynomial equation \( w^3 + 6w^2 + 8w = 0 \).
For \( w = 0 \): \[ 0^3 + 6 \times 0^2 + 8 \times 0 = 0 \]
- The equation holds true.
For \( w = -2 \): \[ (-2)^3 + 6 \times (-2)^2 + 8 \times (-2) = -8 + 24 - 16 = 0 \]
- The equation holds true.
For \( w = -4 \): \[ (-4)^3 + 6 \times (-4)^2 + 8 \times (-4) = -64 + 96 - 32 = 0 \]
- The equation holds true.

Since all solutions satisfy the original equation, we have verified our solutions correctly!