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Equation Solving
FOIL is the sixth 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 - FOIL - Important for this Puzzle
”FOIL” is an acronym for the words “first, outside, inside, last” and is a memory aid for remembering the steps by which two binomials are multiplied together. More to follow below!
FOIL - the Cache
Some Definitions
Binomial: We learned what the words “term” and “expression” meant - and they still do! - in Brackets. A binomial is an expression of two terms - for example, 2m - 3n.
Trinomial: A trinomial is an expression of three terms - for example, 4p + 6q - 2r.
Polynomial: Remember “polygon” from Decagon? A polynomial is a general word, referring to any expression with several terms, for example, 7a + 3b - 4c + 8d + 5e - 9f . A trinomial is a polynomial too but with a specific number of terms.
Power: A simple example of a power is x2 where the “x” is referred to as the base and the “2" is referred to as the exponent. Other examples of powers would be: 3bm - here, the “3" is not part of the power - just a numerical coefficient; (4t)5 - here, the entire “4t”, being in brackets, is the base of the power so the exponent applies to both the “4" and the “t.” (This might be a good time to mention that an “x” or “y” or “r” is considered to be a "power" too, the exponent being understood to be 1.)
Multiplying Powers
Of Different Bases: An example will suffice: x2 multiplied by y4 is simply x2y4.
Of the Same Base: Consider what x3 means: it means x times x times x. Consider what x5 means: it means x times x times x times x times x.
So what will (x3)(x5) mean?
It means (x times x times x)(x times x times x times x times x) which, in turn, means
x times x times x times x times x times x times x times x or x8.
So, to multiply two - or more - powers of the same base, just use that base and add the exponents.
Multiplying Powers Having Numerical Exponents: Here, again, an illustration or two will help:
(1) (3x4)(2x5)
= 6x9
(2) (5a2b3)(6a4b2)
= 30a6b5.
Simply multiply the numerical coefficients and deal with the powers as shown above.
Multiplying Two Binomials (FOIL)
What Does FOIL Mean?
Suppose you were asked to multiply (2a + 5) by (3a + 4)
FOIL dictates that we multiply in order:
- the two First terms;
- the two Outside terms;
- the two Inside terms;
- the two Last terms.
Why Does it Work?
Let’s consider a numerical example first. We know that (6)(11) = 66.
But we could write that as two binomials, (2 + 4)(8 + 3). So let us use the steps of FOIL to do the calculation:
(2 + 4)(8 + 3)
= F(2)(8) + O(2)(3) + I(4)(8) + L(4)(3)
= 16 + 6 + 32 + 12
= 66. And that's how FOIL works.
So, (2a + 5)(3a + 4)
= 6a2 + 8a + 15a + 20
= 6a2 + 23a + 20 [We simplify by adding like terms.]
More Examples
(By the way, please note that if there is more than one letter in a term, we usually write them in alphabetical order, that is, “ab” instead of “ba.”)
(1) (a + b)(x + y)
= ax + ay + bx + by
(2) (2a + 5b)(4a + 2b)
= 8a2 + 4ab + 20ab + 10b2
= 8a2 + 24ab + 10b2
(3) (4x - 3y)(2x + 5y)
= 8x2 + 20xy - 6xy - 15y2
= 8x2 + 14xy - 15y2.
(4) (3p2 + 5q4)(2p6 - 4q3
= 6p8 - 12p2q3 + 10p6q4 - 20q7.
Now Your Questions!
A: Simplify Each Expression (if necessary) and Provide the Requested Numbers
(1) The expression (2a - 3b) is a:
- if it is a monomial, A = 2;
- if it is a binomial, A = 4;
A = ___.
(2) In the power, 35,
- the base is ___ (B);
- the exponent is ___. (C)
(3) Multiply and simplify if possible:
(2p + 5q)(3p - 4q).
In the most simplified answer,
- the number of terms is ___ (D);
- the numerical coefficient of the pq term is ___ (E).
(4)Multiply and simplify if possible:
(7x2 + 2y3)(x4 - 5y2).
In the most simplified answer,
- the largest exponent in the answer is ___ (F);
- the number of terms in the answer is ___. (G)
B: Simplify and Solve Each Equation by Inspection (see Abacus if necessary)
(1) (x + 2)(x + 5) - (x2 + 7x + 6) = (x - 3)(x + 4) - (x2 - 9) Call the answer H.
(Please refer to Ellipse if necessary for dealing with the implied "-1" in front of the brackets.)
(2) (4x - 1)(x + 3) - 2(2x2 + 5x) = (2x + 5)(2x - 3) - 4(x2 + x - 4) Call the answer J.
Summary of Answers Above
A = ____ ; B = ___ ; C = ___ ; D = ___ ; E = ___ ; F = ___ ; G = ___ ;
H - the answer to B(1) = ____ ; J - the answer to B(2) = ___ .
The Co-ordinates of the Cache
Calculations
The co-ordinates are:
N 44° a b . c d e’ and W 078° f g . h j k’ where these lower case letters are not related to the upper case letter values except as specified below.
a = C - B = ___ ; b = E - D = ___ ; c = H - (E - F) = ___ ; d = A + F - G = ___ ;
e = J + B = ___ ; f = E - D = ___ ; g = H - (E - C) = ___ ; h = C + G = ___ ;
j = F - C = ___ ; k = (A + G) - (B + D - J) = ___.
The Co-ordinates
N 44° ___ ___ . ___ ___ ___ ‘ and W 078° ___ ___ . ___ ___ ___ ‘.
Additional Comments
- Please bring your own writing utensil;
- Hours of operation in this area: May to October: 07:00 - 21:00; November to April: 08:00 - 18:00;
- Note the parking areas provided - one in the “winter” months, three more in the "summer" months;
- 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;
- Co-ordinates are "swingy" in here so be patient with your device - it's doing its best;
- Be aware of the possibility of others on the trail;
- Please feel free to confirm your answer on GeoChecker.com.