Math Enrichment Program at Don Bosco CSS
January 31, 2011
Equations I
Part 0: Warm-up Activities
 Birthday Puzzle
 Riding Horses
Part 1: Theoretical Concept – Definitions & Theorems
Definitions
Linear Equation: equation in which all the variables appear to the first power (exponent one)
Ordered n-tuple: a solution to a learn equation, of real numbers which satisfy the linear equation
ℝ𝑛 : set of all ordered n-tuples with real entries.
Linear System of Equations: a set of one or more linear equations taken together.
Particular Solution to A Linear System of Equations: an ordered n-tuple, (pair, triple, quadruple,. . . ), of
real numbers which satisfies all of the linear equations in the system. To find particular solution you
have to give values to all but one of the variables and calculate the value of that variable. Pythagorean
Triplet: solution (x,y,z) of the equation x2 + y2 = z2
By Xiaoxiao Zhao
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Math Enrichment Program at Don Bosco CSS
January 31, 2011
General Solution of A Linear System of Equations/Parametric Solutions: the set of all solutions to the
system of linear equations. These solutions will involve a parameter, or parameter.
Exponential Functions: A function that can be expressed in the form 𝑓(𝑥) = 𝑎 ∙ 𝑏 𝑥 , 𝑎 ≠ 0, 𝑏 ≠ 1, and b
is positive.
Theorems
Fermat’s Last Theorem
There are no non-zero solutions to the equations xn + yn = zn for n ∈ 𝑁, 𝑛 ≥ 3
The Factor Theorem
If we find a root of f (x), say 𝑟1, then (x - 𝑟1 ) is a factor of f (x). In fact the Theorem says more: each
root corresponds to one factor and each factor corresponds to one root.
The Remainder Theorem
Given polynomials f (x) and (x -r), we can write f (x) = (x - r) g(x) + R(x), where g(x) is the quotient and
R(x) is the remainder.
Rational Root Theorem
A constraint on rational solutions (or roots) of the polynomial equation
𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−1 𝑥 𝑛−2 + … + 𝑎1 𝑥 + 𝑎0 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑎𝑖 ∈ ℤ
If a0 and an are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms
(i.e., the greatest common divisor of p and q is 1), satisfies
 p is an integer factor of the constant term a0
 q is an integer factor of the leading coefficient an.
𝑝
Thus, a list of possible rational roots of the equation can be derived using the formula 𝑥 = ± 𝑞.
Fundamental Theorem of Algebra
Given any polynomial
𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−1 𝑥 𝑛−2 + … + 𝑎1 𝑥 + 𝑎0 , 𝑎𝑖 ∈ ℚ
we know it has n roots, all complex numbers. We can always write
𝑓(𝑥) = (𝑥 − 𝑟1 )(𝑥 − 𝑟2 ) … (𝑥 − 𝑟𝑛 ) , where 𝑟1 , 𝑟2 , … , 𝑟𝑛 are the n complex roots of f (x)
By Xiaoxiao Zhao
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Math Enrichment Program at Don Bosco CSS
January 31, 2011
Part 2: Exercise
1. Determine a few particular solutions to the equations, and then determine a general solution:
1.1 Equations in ℝ2
a) x = 3
b) x + y = 3
1
c) 2 𝑥 − 3𝑦 = −4
1.2 Equations in ℝ3
d) x + y + z = 3
e) y = 3
1.3 Equations in ℝ4
f) w + 2x + 3y + 4z = 42
2. Are there non-zero rational numbers x; y; z such that x2 + y2 = z2. If so, name some.
By Xiaoxiao Zhao
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Math Enrichment Program at Don Bosco CSS
January 31, 2011
3. Are there non-zero rational numbers x; y; z such that x3 + y3 = z3? If so, name some.
(Hint: Fermat’s Last Theorem)
4. Derive the Quadratic Formula by completing the square on ax2 + bx + c = 0. How many roots does
this formula give?
5. Give the Cubic Formula:
and a cubic equation f (x) = 3x3 - 2x3 + 4x-5. If we solve it, we will end up with 3 roots. Besides
using the Cubic Formula to solve this equation, is there any easier way to help us find solutions?
(Hint: The Factor Theorem & he Remainder Theorem)
By Xiaoxiao Zhao
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Math Enrichment Program at Don Bosco CSS
January 31, 2011
6. Let f (x) = x4 + 5x3 + 5x2 - 5x – 6. Solve for the roots of f (x) by factoring f (x) completely.
7. Find x such that 2x= 5. (Exponential Functions)
2
8. Solve the following for x: 10𝑥 = 10, 000 (Exponential Functions)
𝑥
𝑥
*How would you go about solving 77 = 93 𝑎𝑛𝑑 7𝑥 = 93 ?
By Xiaoxiao Zhao
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Math Enrichment Program at Don Bosco CSS
January 31, 2011
For more resources and the electronic copy of this handout, please visit our website at
http://mathenrichmentdbcss.wikispaces.com/
The next Enrichment Program Meeting will be on February 7th at 3 pm in Resource Room.
For any concerns and questions regarding today’s meeting, feel free to post them on the
discussion board of our website.
References
The Centre for Education in Mathematics and Computing It is the website from University of Waterloo. You
can find a wide variety of math enrichment resources including Past Waterloo Math Contest, Enrichment
Workshop (problem sets & online-lecture), Careers related to Mathematics, Other Web Resources for Students
at Different Grade Levels.
By Xiaoxiao Zhao
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