Problem 1: Simplify
x^8+8x^6-729x^2-5832 x^3+1728
x^2+9 x^5+8x^3+1728x^2+13824
This problem takes a lot of factoring. You have to know how to factor squares and cubes, along with knowing how to factor by grouping. First, factor the x^8+8x^6-729x^2-5832. Pulling out the x^6 in the first pair results in (x^2+8) which happens to the other pair when you pull out 729. You should end up with (x^2+8)(x^6-729). Using the rule of squares, (x^6-729) is simply (x^3+27)(x^3-27), this can also be factored by the rule of thirds. (x^3+27) factors to (x+3)(x^2-3x+9) and (x^3-27) factors to (x-3)(x^2+3x+9). That's it for now, the x^2+9 factors to (x+3)(x-3) and x^3+1728 factors to (x+12)(x^2-12x+144). The last tricky one is the x^5+8x^3+1728x^2+13824. By pulling out x^3 by grouping you get a factor of (x^2+8) and by pulling out 1728 from the second pair you get the same factor, resulting in (x^2+8)(x^3+1728), and the (x^3+1728) looks familiar. Factor it again to get (x+12)(x^2-12x+144). What you should have by now should look like this...
(x^2+8)(x+3)(x-3)(x^2-3x+9)(x^2+3x+9)(x+12)(x^2-12x+144)
(x+3)(x-3)(x^2+8)(x+12)(x^2-12x+144)
Cross out the matching pairs...
And you get...
(x^2-3x+9)(x^2+3x+9)!
Problem 2: Simplify
x^2-4 x^3-2x^2-4x+8
x^5-3x^3+27x^2-81 x^5-3x^3-8x^2+24
Lots o' factoring! First though, you have to flip the fraction because we are dividing...
x^2-4 x^5-3x^3-8x^2+24
x^5-3x^3+27x^2-81 x^3-2x^2-4x+8
Now we can start. That simple x^2-4 is just (x+2)(x-2). Using grouping and pulling out x^2-3 from the pairs in x^5-3x^3-8x^2+24, you get (x^2-3)(x^3-8) or (x^2-3)(x-2)(x^2+2x+4). Again with grouping x^5-3x^3+27x^2-81, you can pull out x^2-3, making (x^2-3)(x^3+27) or (x^2-3)(x^2-3x+9). Lastly x^3-2x^2-4x+8 can group pulling out x-2, making that (x-2)(x^2-4) or (x-2)(x-2)(x+2). So you should have...
(x-2)(x+2)(x^2-3)(x-2)(x^2+2x+4)
(x^2-3)(x+3)(x^2-3x+9)(x-2)(x-2)(x+2)
Cross out pairs...
And [musical instrument] you get...
x^2+2x+4
(x+3)(x^2-3x+9)
Problem 3:
Your uncle is a kind man. A kind man with a Math major. When you were born, out of the goodness of his heart, he put $100 in your banking account to increase quarterly. Every birthday he presented you with a problem saying, "If you can find the rate at which your bank account grows, I'll let you take the money and use it." On your 18th birthday, worried about college on the horizon he give you the solution to the rate, in problem form...
4x+5 25 7
x^2 5x^2 5x^2
In order to con your uncle out of the money, each of the fraction must be equal. Multiplying the first fraction by 5 makes all of them equal...
20x+25 25 7
5x^2 5x^2 5x^2
Now multiply all of the factions by 5x^2 gives you...
20x+25-25=7
An easy to solve equation. 25-25 gets rid of both leaving 20x=7. Divide 7 by 20 and the rate is...
7
20
Problem 4:
Continuing on with your uncle you tell him the rate is 7/20. He congratulates you, however, he says, "Now in order to get the money, tell me how much you've made." You remember back, he put $100 in, increasing quarterly, at a rate of 7/20, for 18 years. Something like...
x = 100(1+.35/4)^(4*18)
Well, 4*18= 72 and 1 + .35/4= 1.0875. So 1.0875^72= 419.6692. That times 100...
x = $41966.92!
That's a good start for college!
Reflection:
I choose the concepts in these problems because I enjoy seeing how everything works out and in an even amount. I like knowing what each number means, along with grouping and the rules of squares and cubes. Seeing fractions solve out and compounding equations are also fun for me.
These problems provide an overview of my understanding of what I have learned so far, simply because I did not know them before. They make sense, but it is interesting to see something that I know contains more to be learned. It's like taking a trip to a famous spot and going back again, expecting to see the same sight, but instead finding what may have been overlooked the first time.
I learned from making these problems, that making a challenging problem, in itself, is challenging! I don't get how teachers do it so easily. A teacher may look at my problems and already know the answers, even after my hours of effort trying to make them hard. I'm thankful for this project.
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