Assignment #1: The 26th
What is (x-a)(x-b)(x-c)...(x-y)(x-z)? Explain how you found the answer.
In this problem, I am going to
have to multiply 26 binomials together.
Solve/Investigate the Problem
My initial plan was to multiply
each binomial one by one, using the distributive property, until I reached
the last term in the sequence.
of the Problem
For the first terms in the
problem, I utilized the mnemonic, FOIL, to distribute my terms through the
binomial. After figuring that
multiplying each binomial individually was a tedious process, I looked for
short cuts to solving the problem.
I did notice a pattern of multiplying the binomials, which helped me
to determine the first and last term in the sequence. The first term in the sequence was always
x, followed by an exponent representative of its place in the
alphabet. So, after using the
distributive property the first time, the first term was x². After performing the distributive
property for the third time, the term was x³, and so on. I used my knowledge of positive and
negative integers to determine that the last term in the sequence would be
positive. I also noticed that
the last term in the sequence would be the product of all the letters
together. For example, abcdef…..xyz. As a group, we created a chart that
showed the patterns of terms after multiplying each successive
binomial. Multiplying the first
two binomials would produce 4 terms or 2². Multiplying the next binomial would
produce 8 terms or 2³, and so on.
This process would have produced 2terms, which would have been too large to determine. So, I had to look for an alternative
to solve the problem. After much
thought, I determined that one of the binomials in the sequence was (x-x),
which results in an answer of 0.
According to the multiplicative property of zero, anything
multiplied by 0 is 0. Thus, the
answer is 0.
Extensions of the Problem
There were many extensions in
this problem, which were mentioned in my investigation. Other areas discussed include quadrants,
coordinate system, and quadratic equations. More specifically, we talked about
the four quadrants created by the coordinate plane and how it may relate to
the naming of the quadratic equation.
This led into a discussion of parabolas, and how to graph solutions
to a set.
Author & Contact
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