https://file.aiquickdraw.com/mathbot/file/1726670519331utdi0rtd.jpg

The problem asks whether the statement "Regardless of what real numbers I choose for \(x\) and \(y\), I will always obtain zero when evaluating \(2x^2y - 2yx^2\)" makes sense.

Let's break this down:

### Expression:
\[
2x^2y - 2yx^2
\]

Rewriting this expression by factoring out the common terms:
\[
2x^2y - 2yx^2 = 2x^2y - 2x^2y = 0
\]

This simplifies to zero regardless of the values of \(x\) and \(y\), since both terms are identical and cancel each other out.

Now, let's analyze the answer choices based on this:

1. **Choice A**: "The statement does not make sense because \(2x^2y = 2yx^2\), which is not always equal to 0 for any real numbers \(x\) and \(y\)."
   - This is incorrect because the expression **always** simplifies to zero, so the statement **does** make sense.

2. **Choice B**: "The statement makes sense because terms with the \(2x^2y\) and \(2yx^2\) factors can be combined, and therefore, \(2x^2y - 2yx^2 = 0\)."
   - This is the correct answer. The terms indeed cancel each other out, leading to zero.

3. **Choice C**: "The statement does not make sense because \(2x^2y = 2yx^2 = x\), which is not always equal to 0 for any real numbers \(x\) and \(y\)."
   - This is incorrect because \(2x^2y - 2yx^2\) always equals zero, not \(x\).

4. **Choice D**: "The statement does not make sense because \(2x^2y - 2yx^2 = xy\), which is not always equal to 0 for any real numbers \(x\) and \(y\)."
   - This is also incorrect. The expression simplifies to zero, not \(xy\).

Thus, the correct answer is **B**.

---

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. Can any algebraic expression be simplified by factoring?
2. Why is it important to factor out common terms in expressions?
3. How can you recognize when terms in an expression cancel each other out?
4. Does this simplification technique work for all polynomials?
5. What other techniques can you use to simplify expressions?

**Tip**: Always look for common factors when simplifying algebraic expressions to spot potential simplifications.

Determine whether the statement makes sense or does not make sense, and explain your reasoning. Regardless of what real numbers I choose for x and y, I will always obtain zero when evaluating 2x^2y - 2yx^2.

Understand how to simplify the polynomial 2x^2y - 2yx^2 and determine whether it always results in zero for any real values of x and y. This algebra problem covers key concepts like factoring and the commutative property of multiplication. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation