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If two of the four expressions \(\mathrm{x + y}\), \(\mathrm{x + 5y}\), \(\mathrm{x - y}\), and \(\mathrm{5x - y}\) are chosen at random, what is the probability that their product will be of the form of \(\mathrm{x^2 - (by)^2}\), where b is an integer?
Let's start by understanding what we're looking for in everyday terms. We have four algebraic expressions, and we need to pick any two of them randomly. Then we multiply those two expressions together and see if the result looks like \(\mathrm{x^2 - (by)^2}\), where b is some whole number.
The form \(\mathrm{x^2 - (by)^2}\) is called a "difference of squares" - it's when you have one perfect square minus another perfect square. For example, \(\mathrm{x^2 - 4y^2}\) or \(\mathrm{x^2 - 25y^2}\) would fit this pattern.
So our job is to figure out: if we randomly pick 2 expressions from our list of 4, what's the chance that when we multiply them, we get this special difference of squares pattern?
The four expressions we're choosing from are:
Process Skill: TRANSLATE
When we choose 2 expressions from 4 expressions, we need to count how many different ways we can do this. Let's list them out systematically:
Pair 1: \(\mathrm{(x + y)}\) and \(\mathrm{(x + 5y)}\)
Pair 2: \(\mathrm{(x + y)}\) and \(\mathrm{(x - y)}\)
Pair 3: \(\mathrm{(x + y)}\) and \(\mathrm{(5x - y)}\)
Pair 4: \(\mathrm{(x + 5y)}\) and \(\mathrm{(x - y)}\)
Pair 5: \(\mathrm{(x + 5y)}\) and \(\mathrm{(5x - y)}\)
Pair 6: \(\mathrm{(x - y)}\) and \(\mathrm{(5x - y)}\)
So there are 6 total possible pairs we could randomly select. This gives us our denominator for the probability calculation.
Now we need to multiply each pair and see which ones give us the difference of squares pattern \(\mathrm{x^2 - (by)^2}\). Remember, a difference of squares comes from the pattern \(\mathrm{(a + c)(a - c) = a^2 - c^2}\).
Pair 1: \(\mathrm{(x + y)(x + 5y)}\)
\(\mathrm{= x^2 + 5xy + xy + 5y^2}\)
\(\mathrm{= x^2 + 6xy + 5y^2}\)
This is NOT a difference of squares (it has a positive middle term and positive y² term).
Pair 2: \(\mathrm{(x + y)(x - y)}\)
\(\mathrm{= x^2 - xy + xy - y^2}\)
\(\mathrm{= x^2 - y^2}\)
This IS a difference of squares! We can write this as \(\mathrm{x^2 - (1 \cdot y)^2}\), so b = 1.
Pair 3: \(\mathrm{(x + y)(5x - y)}\)
\(\mathrm{= 5x^2 - xy + 5xy - y^2}\)
\(\mathrm{= 5x^2 + 4xy - y^2}\)
This is NOT a difference of squares (the x² coefficient isn't 1, and we have a middle term).
Pair 4: \(\mathrm{(x + 5y)(x - y)}\)
\(\mathrm{= x^2 - xy + 5xy - 5y^2}\)
\(\mathrm{= x^2 + 4xy - 5y^2}\)
This is NOT a difference of squares (it has a positive middle term).
Pair 5: \(\mathrm{(x + 5y)(5x - y)}\)
\(\mathrm{= 5x^2 - xy + 25xy - 5y^2}\)
\(\mathrm{= 5x^2 + 24xy - 5y^2}\)
This is NOT a difference of squares (the x² coefficient isn't 1, and we have a middle term).
Pair 6: \(\mathrm{(x - y)(5x - y)}\)
\(\mathrm{= 5x^2 - xy - 5xy + y^2}\)
\(\mathrm{= 5x^2 - 6xy + y^2}\)
This is NOT a difference of squares (the x² coefficient isn't 1, and we have a middle term).
Process Skill: CONSIDER ALL CASES
From our systematic check, we found that only 1 pair out of the 6 possible pairs produces a difference of squares pattern:
Therefore, the probability = Number of favorable outcomes / Total number of possible outcomes = \(\mathrm{\frac{1}{6}}\)
The probability that two randomly chosen expressions will have a product of the form \(\mathrm{x^2 - (by)^2}\) is \(\mathrm{\frac{1}{6}}\).
The answer is E.
1. Misunderstanding the target form \(\mathrm{x^2 - (by)^2}\)
Students might not recognize that \(\mathrm{x^2 - (by)^2}\) represents a difference of squares pattern where the coefficient of x² must be exactly 1, and the y term must be a perfect square with coefficient b². They may incorrectly think expressions like \(\mathrm{5x^2 - y^2}\) or \(\mathrm{x^2 + y^2}\) qualify as valid forms.
2. Incorrect counting of total possible pairs
Students may confuse combinations with permutations and think there are 8 possible ways to choose pairs (counting order), or use the wrong combination formula. Some might incorrectly count 4 pairs by missing that we're selecting 2 from 4 expressions, not pairing each with the next one sequentially.
3. Misinterpreting "chosen at random"
Students might think the question is asking about choosing expressions with replacement or in a specific order, rather than understanding this as a straightforward combination problem where we select 2 different expressions from the 4 given.
1. Algebraic multiplication errors
When expanding products like \(\mathrm{(x + 5y)(5x - y)}\), students commonly make sign errors or forget terms. For example, they might get \(\mathrm{5x^2 - xy + 25xy - 5y^2}\) but incorrectly combine the middle terms as 24xy instead of the correct computation, or mix up positive and negative signs during distribution.
2. Incorrectly identifying difference of squares
Students may mistakenly think expressions like \(\mathrm{x^2 + 4xy - 5y^2}\) qualify as difference of squares because they see x² and -5y², ignoring that the middle term (4xy) disqualifies it. They might also accept forms like \(\mathrm{5x^2 - y^2}\) where the x coefficient isn't 1.
3. Missing the constraint that b must be an integer
Even when students correctly identify \(\mathrm{x^2 - y^2}\) as a difference of squares, they might not verify that b = 1 is indeed an integer, or they might incorrectly accept cases where the y-coefficient doesn't yield an integer value for b.
4. Errors while selecting the answer
No likely faltering points - once students have correctly identified that only 1 out of 6 pairs works, the final probability calculation of \(\mathrm{\frac{1}{6}}\) is straightforward and directly matches answer choice E.