Jill is factoring the expression $13 x y-52 y$. Her work is shown below.
Factors of $13 x y: 1,13, x, y$
Factors of 52y. 1, 2, 26, 52, $y$
GCF: $y$
Factored expression: $y(13 x-52)$
Which best describes the accuracy of Jill's solution?
A. Jill's solution is accurate.
B. Jill omitted a factor pair, which affected the GCF and factored expression.
C. Jill made an error when determining the GCF from her list of factors.
D. Jill made an error when writing the factored expression.
Answer (2)
Jill's factors of 13 x y and 52 y are incomplete.
The greatest common factor (GCF) of 13 x y and 52 y is 13 y , not y .
Factoring 13 x y − 52 y using the correct GCF gives 13 y ( x − 4 ) .
Jill omitted a factor, affecting the GCF and factored expression: Jill omitted a factor pair, which affected the GCF and factored expression.
Explanation
Problem Analysis Let's analyze Jill's factoring of the expression 13 x y − 52 y . We need to identify if she made any errors in finding the factors, the greatest common factor (GCF), or in writing the factored expression.
Factors of 13xy First, let's list the factors of 13 x y . The factors are 1 , 13 , x , y , 13 x , 13 y , x y , 13 x y . Jill's list, 1 , 13 , x , y , is incomplete.
Factors of 52y Next, let's list the factors of 52 y . Since 52 = 2 × 2 × 13 , the factors are 1 , 2 , 4 , 13 , 26 , 52 , y , 2 y , 4 y , 13 y , 26 y , 52 y . Jill's list, 1 , 2 , 26 , 52 , y , is also incomplete.
Finding the GCF Now, let's find the greatest common factor (GCF) of 13 x y and 52 y . The factors of 13 x y are 1 , 13 , x , y , 13 x , 13 y , x y , 13 x y . The factors of 52 y are 1 , 2 , 4 , 13 , 26 , 52 , y , 2 y , 4 y , 13 y , 26 y , 52 y . The common factors are 1 , 13 , y , 13 y . Therefore, the GCF is 13 y . Jill incorrectly identified the GCF as y .
Factoring the Expression Now, let's factor the expression 13 x y − 52 y using the correct GCF, 13 y . Factoring out 13 y from both terms, we get 13 y ( x − 4 ) . Jill's factored expression is y ( 13 x − 52 ) . We can see that Jill's expression is not completely factored since 13 x − 52 = 13 ( x − 4 ) .
Accuracy of Jill's Solution Comparing Jill's solution to the correct solution, we see that Jill omitted a factor pair, which affected the GCF and the factored expression.
Conclusion Therefore, the best description of the accuracy of Jill's solution is that Jill omitted a factor pair, which affected the GCF and factored expression.
Examples
Factoring expressions is a fundamental skill in algebra and is used in many real-world applications. For example, if you are designing a rectangular garden with an area of 13 x y − 52 y square feet, where x and y are variables representing the dimensions, factoring the expression to 13 y ( x − 4 ) helps you understand the possible dimensions of the garden. You know that one side must be 13 y feet long and the other side must be ( x − 4 ) feet long. This allows you to plan the layout and fencing requirements more effectively.
Jill's solution to factoring 13 x y − 52 y is inaccurate due to her incomplete identification of factors and incorrect GCF. The correct GCF is 13 y , leading to the factored expression 13 y ( x − 4 ) . Therefore, the best choice is B, as her omission of a factor pair affected her work.
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