Question

A student factored 10 x Superscript 5 Baseline y squared plus 2 xy as 2 xy left parenthesis 5 x Superscript 4 Baseline y right parenthesis. What went​ wrong? Factor correctly.

Answer 1

Let's factor the expression \(10x^5y^2 + 2xy\) correctly.

1. **Identify the greatest common factor (GCF)**:
- The coefficients are \(10\) and \(2\). The GCF of these coefficients is \(2\).
- For the variables, the GCF of \(x^5\) and \(x^1\) is \(x^1 = x\).
- For \(y^2\) and \(y^0\) (since \(2xy\) doesn't have a \(y\) term, we consider it as \(y^0\)), the GCF is \(y^0 = 1\).

Thus, the overall GCF of the expression \(10x^5y^2 + 2xy\) is \(2xy\).

2. **Factor out the GCF**:
\[
10x^5y^2 + 2xy = 2xy(5x^4y + 1)
\]

So the correct factored form of the expression is:

\[
2xy(5x^4y + 1)
\]

### Explanation of the student's error:
The student incorrectly factored out \(2xy\) and then stated \(5x^4y\) in the remaining factor without accounting for the correct term. The correct remaining term when factoring \(2xy\) from \(10x^5y^2 + 2xy\) is \(5x^4y + 1\).

Thus, the correctly factored expression is \(2xy(5x^4y + 1)\).