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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.

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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The common factor for the​ coefficients, 2​, is not correct. The correct factored form is 10 x Superscript 5 Baseline y squared plus 2 xy equals nothing .
  
B.
The common factors for the variable terms x and y are not correct. The correct factored form is 10 x Superscript 5 Baseline y squared plus 2 xy equals nothing .
  
C.
The expression inside the parentheses is not correct. The correct factored form is 10 x Superscript 5 Baseline y squared plus 2 xy equals nothing .
  
D.
The factor 2xy is not a common factor. The correct factored form is 10 x Superscript 5 Baseline y squared plus 2 xy equals nothing .

1 answer

To analyze the student's factoring of the expression \(10x^5y^2 + 2xy\), we can first identify the greatest common factor (GCF) in the expression.

  1. Identify the GCF of the coefficients:

    • The coefficients are 10 and 2. The GCF is 2.

  2. Identify the GCF of the variable terms:

    • For the \(x\) terms, we have \(x^5\) and \(x\). The GCF is \(x\) (the lowest power).
    • For the \(y\) terms, we have \(y^2\) and \(y\). The GCF is \(y\) (the lowest power).

Putting it all together, the GCF of the entire expression \(10x^5y^2 + 2xy\) is \(2xy\).

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

Now let's check the provided options:

A. The common factor for the coefficients, 2, is not correct. The correct factored form is \(10x^5y^2 + 2xy = _ \).

B. The common factors for the variable terms \(x\) and \(y\) are not correct. The correct factored form is \(10x^5y^2 + 2xy = _ \).

C. The expression inside the parentheses is not correct. The correct factored form is \(10x^5y^2 + 2xy = _ \).

D. The factor \(2xy\) is not a common factor. The correct factored form is \(10x^5y^2 + 2xy = _ \).

Analysis of the options:

  • The student's factorization was incorrect. They obtained \(2xy(5x^4y)\), which is incorrect.
  • The correct factorization we found is \(2xy(5x^4y + 1)\), indicating that the factor inside the parentheses was incorrect.

The most appropriate choice is:

C. The expression inside the parentheses is not correct. The correct factored form is \(10x^5y^2 + 2xy = 2xy(5x^4y + 1)\).