https://file.aiquickdraw.com/mathbot/file/1726380859004zj7cvv4n.jpg

Let’s go through the first few problems of the "Concept Check" section from the image.

### 1. Factored Form of \( x^3 - x^2 - 6x = x(x - 3)(x + 2) \)
This equation can be solved by setting each factor to zero:
\[
x(x - 3)(x + 2) = 0
\]
- \( x = 0 \)
- \( x - 3 = 0 \Rightarrow x = 3 \)
- \( x + 2 = 0 \Rightarrow x = -2 \)

Thus, the solutions to \( x^3 - x^2 - 6x = 0 \) are:
\[
x = 0, 3, -2
\]

### 2. What value of \( x \) is not in the domain of the equation \( \frac{1}{x - 5} + 3 = \frac{x + 1}{x - 5} \)?
For this rational equation, the denominator \( x - 5 \) cannot be zero. Therefore, \( x = 5 \) is not in the domain of the equation.

### 3. Clearing Fractions in Exercise 2
To clear the fractions in the equation from Exercise 2:
\[
\frac{1}{x - 5} + 3 = \frac{x + 1}{x - 5}
\]
Multiply every term by \( x - 5 \) to eliminate the denominators.

### 4. Solutions to \( \sqrt{x + 7} - x = 5 \)
A student found \( x = -6 \) and \( x = -3 \). To determine which is extraneous, substitute each solution back into the original equation to check if both sides are equal. I can help further if needed with that verification.

### 5. Solving \( x^{2/3} = 9 \)
To solve \( x^{2/3} = 9 \), raise both sides of the equation to the power of \( 3/2 \), which is the reciprocal of \( 2/3 \).

Let me know if you'd like further details or if you'd like to proceed with other problems.

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Here are some related questions:
1. How do you solve cubic equations using the factorization method?
2. What does it mean for a value to be "not in the domain" of a function?
3. Why do we multiply both sides by a common denominator to clear fractions?
4. What is an extraneous solution in radical equations?
5. How do you handle fractional exponents when solving equations?

**Tip:** Always verify potential solutions by substituting them back into the original equation, especially when dealing with radicals or rational expressions.

The factored form of the polynomial x^3 - x^2 - 6x is x(x - 3)(x + 2). What are the solutions of the equation x^3 - x^2 - 6x = 0?

Explore how to solve the cubic polynomial equation x^3 - x^2 - 6x = 0 using factoring techniques. This step-by-step solution applies key algebraic concepts such as factoring and the zero product property. Suitable for students in Grades 9-10.

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