The “spring check” in 2019 will be the last of the regular formal checks at this location; hereafter, we will visit only when we are informed in a log or personal e-mail that maintenance is needed - so please keep us informed!
Equation Solving
Googol is the seventh in a series of fifteen puzzle caches of increasing difficulty based on solving equations. It is recommended that one start with Abacus (GC6A006) and continue in alphabetical sequence through Brackets (GC6AR81), Compass (GC6AWD0), Decagon (GC6C4EE), Ellipse (GC6CFW3), FOIL (GC6HDNE), Googol (GC751JK), Hectare (GC75BD3), Index (GC75EAM), Jargon (GC75X8W), Kite (GC76010), LessThan (GC763HV ), Median (GC7DH17), Nonagon (GC7DWC4) and Obelus (GC7DWCD).
The Name - Googol - Irrelevant for this Puzzle
“Googol” is a word which one dictionary we checked stated was “not in formal use” and a second that it was a “fanciful word, not found in technical use.” One reason for its lack of use is probably that it stands for a number so large that it is really beyond our comprehension: 10100 or a one with a hundred zeros after it. Just to give an idea of how big a number googol is, one source says that if you multiplied the mass of an electron - about 0.000 000 000 000 000 000 000 000 000 91 g - by googol, you would have a mass larger than the mass of the visible universe! Enough! Let’s move on!
Googol - the Cache
Multiplying Polynomials
The Process
As we learned in FOIL, a polynomial is an expression with several terms in it. To multiply two polynomials, one must simply ensure that one multiplies each term in the first polynomial by every term in the second polynomial - as the FOIL process ensures is done with two binomials. And remember that brackets together - as in (3)(4) - mean to multiply. Let us illustrate by two examples - one numerical and one with those nasty letters!
First, to simplify (7 + 2 - 5)(4 - 1 + 6) we would normally simplify in the brackets as follows :
(7 + 2 - 5)(4 - 1 + 6)
= (4)(9)
= 36.
However, suppose we couldn’t simplify within the brackets as in the case with, for example, (3x - 2y + 5); we would then do the above example in this way:
(7 + 2 - 5)(4 - 1 + 6)
= (+7)(+4) + (+7)(-1) + (+7)(+6) . . . [that covers the multiplying by the +7 at the beginning of the first polynomial]
+ (+2)(+4) + (+2)(-1) + (+2)(+6) . . . [that covers the multiplying by the +2 in the middle of the first polynomial]
+ (-5)(+4) + (-5)(-1) + (-5)(+6) . . . [that covers the multiplying by the -5 at the end of the first polynomial]
= (+28) + (-7) + (+42) + (+8) + (-2) + (+12) + (-20) + (+5) + (-30)
= 28 - 7 + 42 + 8 - 2 + 12 - 20 + 5 - 30
= 28 + 42 + 8 + 12 + 5 - 7 - 2 - 20 - 30
= 95 - 59
= 36 (as before). So . . . it works!
In this second illustration, we use the same process for these two trinomials (expressions with three terms):
(3x2 + 4x - 5)(x2 - 6x + 7)
= 3x4 - 18x3 + 21x2 + 4x3 - 24x2 + 28x - 5x2 + 30x - 35
= 3x4 - 14x3 - 8x2 + 58x - 35. (And that’s as far as one can go!)
Try These to Ensure that You Understand
(1)
(3x + 2)(4x2 - 5x + 6)
(2)
(4a2 - 3a + 5)(a2 + 2a - 7).
(The answers for these should be: 12x3 - 7x2 + 8x + 12 and 4a4 + 5a3 - 29a2 + 31a - 35.)
Now Your Questions!
Simplify Each Expression and Provide the Requested Numbers
(1)
(4p2 - p + 5)(3p2 + 2p - 1)
What is the exponent of the term having 12 as the numerical coefficient? A = ____.
What is the numerical coefficient of the p3 term? B = ____.
What is the exponent of the term having 9 as the numerical coefficient? C = ____.
What is the numerical coefficient of the p term? DE = ____ ____.
(2)
(m2 - 5m + 7)(4m3 - m2 + 2m - 1)
What is the numerical coefficient of the m5 term? F = ____.
What is the exponent of the term having -21 as the numerical coefficient? G = ____.
What is the numerical coefficient of the m3 term? HJ = ____ ____.
What is the exponent of the term having -18 as the numerical coefficient? K = ____.
What is the numerical coefficient of the m term? LM = ____ ____.
And Now, Calculate the Co-ordinates
A = ___ , B = ___ , C = ___ , D = ___ , E = ___ , F = ___ ,
G = ___ , H ___ , J = ___ , K = ___ , L = ___ , M = ___ .
The co-ordinates of the cache are N 44° ab.cde’ and W 078° fg.hij’ where none of the lower case letters is related to the upper case letters above except as defined below:
a = M - (A + H) = ___ ; b = (C)(K) = ___ ; c = (F + G) ÷ A = ___ ;
d = E(D + L) = ___ ; e = M - B ÷ J = ___ ; f = J - C + L = ___ ;
g = D + B = ___ ; h = H + E = ___; i = G ÷ F = ___ ;
j = K(J - H) = ___ .
You Will Find the Cache at:
N 44°__ __ . __ __ __’ and W 078° __ __ . __ __ __’
Additional Comments
- The GPS devices have trouble in here!! This location was marked over 500 trials, but . . . ;
- Please bring your own writing utensil;
- The representatives of the "area" ask that you please stay on the trails until you are as close as you can get to the cache without leaving them;
- This trail has seasonal muddy areas;
- Having referred to "this trail" we should add that there is more than one route to this cache;
- You are looking for a medium-sized "used-to-be" ASA container;
- Note the parking areas provided - one in the “winter” months, three more in the summer months; parking for the whole area costs $2 in any coins; it's your decision as to which one to use!
- Be aware of the possibility of others on the trails;
- Hours of operation in this area: May to October: 07:00 - 21:00; November to April: 08:00 - 18:00;
- There is a parking machine near the permanent parking - $4 for the day; however one might wish first to visit the office - 80 m away - and purchase an annual pass to all "Kawartha Conservation conservation areas" for $84.75 - 20% less for seniors;
- Please feel free to check your answer on GeoChecker.com.