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Foundation Mathematics: Chp 2 - Algebra - Coggle Diagram
Foundation Mathematics: Chp 2 - Algebra
Module 6 - Polynomials and the Binomial Theorem
Expansions
The coefficients of the expansions
(a+b)^n
Can be done manually for smaller
powers by using the FOIL method but
it's much more laborious for larger powers
Second option is to construct Pascals's Triangle (see notes)
Third option is to use the Binomial formula
Pascals triangle
gives us the coefficients
in the expansion of \( (a+b)^n\) = \( (ax^n-1+b^1)\)
Note: the powers of a and b add up to n
Row 0 n = 1
Row 1 n = 2
etc...
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
etc...
The properties somehow lend themselves to finding out the coefficients of the expansion to any n degree
Example of expansion using Pascal's triangle:
\((x+1)^5\)
a) \(1\times x^5\times1^0\) + \((5\times x^4\times 1^1)\) + (10×x^3×1^2) + (10×x^2×1^3) + (5×x^1×1^4) + (1×x^0×1^5)
=
\(x^5 + 5x^4 + 10x^3 +10x^2 + 5x + 1\)
Binomial formula
Factorials
Example: 3! can be written as 3.2.1 and is referred to as 3 factorial
This can be generalised to:
n! = n × (n−1)!
E.g. 3! = 3×(2)! and 2! = 2×(1)! and 1! = 1(0)!
Note: 0! = 1
0! =1. Why?
Let n = 1
then (1-1)! =
Miscellaneous Exercises
14.
To find the values of a + b + c for a parabola
y = ax^2 + bx + c passing through some ordinates (x,y) of say 3 points
a. plug in the values of the 3 points (x,y) into
the quadratic equation to make 3 equations for example
b. Solve for each of a, b and c using simultaneous
equation techniques, re-substitution and elimination
23.
For 2^x = 8^y-1 and 9^y = 3^x-9
Find x and y
Rule 1
: a^f(x)=a^g(x) then f(x)=g(x)
a. Convert 8^y-1 to base 2 ⇒ (2^3)^y-1
b. Therefore, x = 3y-1 and put x in second equation
c. Second equation becomes: 9^y = 3^3(y-1)
d. Use rule one again convert 9^y to base 3 ⇒ (3^2)^y
e. Therefore, 2y = 3y-12 and y=12
f. Substitute y back into x = 3(y-1) = 33
x=33; y=12
Key is converting the bases on left and right hand side to be the same so that rule 1 can be used
28.
Dot product question (multiplication of vectors - the result is a scalar):
Module 5: Algebraic Expressions
Variables
e.g. x + y = 1
The symbols x and y can be assigned any value so they are called variables
Rules of Algebra (common):
Commutativity: The order in which 2 numbers are added or multiplied together does not matter
Associativity: The order in which numbers are grouped together when adding or multiplying does not matter i.e. (2+3)+4 = 2+(3+4)
Distributivity: How operations of multiplication distribute over addition and subtraction (in brackets). Multiplication is left and right distributive (i.e. you arrive at the same answer) but division is only right distributive
Module 7: Solving Algebraic Equations
Unit 3: The Factor Theorem
A polynomial expression written as the product of it's factors like so: x^3-6x^2+11x-6 = 0
If factorised like so: (x-1)(x-2)(x-3) = 0
Then the value of the x variable in which the polynomial equals 0 are called the roots
The Theorem:
So the theorem says that for a polynomial: P =
anx^n+an-1x^n-1...
If (x-a) is a factor of P then x = a (i.e. [a-a] = 0) is a solution to the polynomial so that P = 0
In the above polynomial the solutions are 1,2 and 3
Unit 1: Solving Equations Graphically
A polynomial equated to 0 is called a
polynomial equation
Examples:
1)Linear equation
ax+b = 0. Properties: straight line; where line crosses the x axis is called the solution i.e. where the polynomial equation equals 0
2)Quadratic equation:
ax^2+bx+c = 0. Properties: Shaped like a parabola; where the parabola crosses the x axis is a called the solution; The solutions can be 2,1 or 0 depending on the position of the parabola on the Cartesian coordinate graph system
Module 8: Matrices and Vectors
Unit 4: Geometric Vectors
Geometric vectors are vectors drawn on a Cartesian coordinate system (x and y) e.g. (x,y) beginning point; (x,y)' end point of a certain vector
Also, geometric vectors can be represented with unit vectors i (x-axis) and j (y-axis) so for example: b = 3i+4j
Unit 3: Vectors and their Arithmetric
Module 9: The Right-angled Triangle
Unit 2: Pythagoras' Theorem
The right angled triangle has the following theorem:
a^2 = b^2 + c^2
a being hypotenuse
There are two common proofs:
Algebraic (geometric expressed in algebra)
Geometric
Module 10: Complex Numbers
A complex number is composed of Real and Imaginary numbers:
a + ib; where a and b are real numbers and i^2 = -1 (i being the imaginary number) and √-1 = i
as in 2^2 = 4 and √4 = 2