6A The Language of Polynomials
 A polynomial function is a function that can be written in the form:
P(x) = ax^n + ax^n1 + ... + ax + a
 The leading term, ax^n, of a polynomial is the term of the highest index among those terms with a nonzero coefficient.
The function f(x)=x^1/3
Inverse functions
f(x)=x^{1/3 is the inverse of f(x)= x}3
6E  Cubic Functions of f(x) = a(xh)^3 + k
6E  Cubic Functions of f(x) = a(xh)^3 + k
6E  Cubic Functions of f(x) = a(xh)^3 + k
Point of Inflection (POI) 
Vertical Translations 
Horizontal Translations 
A point of zero gradient 
by adding or subtracting a constant term to yx^3, the graph moves either up or down 
The graph of y=(xh)^3 is moved h units to the right (for h>0) 
The 'flat point' of the graph 
e.g. y=x^3 + k moves the graph k units up (for k>0) 
the POI is at (h,0) 
The POI of graph of y=x^3 is at (0,0) 
The POI becomes (0,k) 
In this case, the graph of y=x^3 is translated h units in the positive direction of the xaxis. 
The implied domain of all cubics is R and the range is also R
6D  Solving Cubic Equations
6D  Solving Cubic Equations
In order to solve a cubic equation, the first step is often to factorise.
Factorise by identifying a factor, then using polynomial division.
Factor the quadratic factor
Then, use null factor law.


6B  Division of Polynomials
 When we divide the polynomial P(x) by the polynomial D(x) we obtain two polynomials, Q(x) the quotient and R(x) the remainder, such that
P(x) = D(x)Q(x) + R(x)
and either R(x) =0 or R9x) has degree less than D(x)
Here P(x) is the dividend and D(x) is the divisor
6B  Division of Polynomials
Dividing Polynomials involving fractions
6B  Division of Polynomials
Dividing polynomials when we have a remainder
6B  Division of Polynomials
Equating Coefficients Methods instead of dividing
6F  Graphs of factorised cubic functions
6F  Graphs of factorised cubic functions
6F  Graphs of factorised cubic functions
Cubic equations with one x intercept
6G  Solving Cubic Inequalities
6H  Families of cubic polynomial functions
y=ax^3 
y=a(xh)^3 + k 
y=a(xa)(xb)(xc) 
y=ax^3+bx^2+cx+d 
6H  Families of cubic polynomial functions
6H  Families of cubic polynomial functions
6B  Division of Polynomials
Equating Coefficients Methods instead of dividing


6C  Special Cases Differences of Cubes
6C  Rational Root Theorem
6C  Rational Root Theorem
P(x) = 2x^{3  x}2  x  3 
Choose factors of 3, which are ±1 and ±3. 
However, P(1) ≠ 0, P(1) ≠ 0, P(3) ≠ 0, and P(3) ≠ 0 
Therefore, we must use the Rational Root Theorem. 
We must use P(± factors of constant/factors of leading coefficient) 
factors of constant (of 3) = ±1, ±3 factors leading coefficient (of 2) = ±1, ±2 
e.g. P(±3/2), P(±1/2) 
We now have to check these factors > P(3/2) = 2(3/2)^{3  (3/2)}2 (3/2) 3 = 0 
therefore, x3/2, which equates to 2x3 as a factor 
6C  Factorisation of Polynomials
For a polynomial P(x)
 If P(a) = 0, then xa is a factor of P(x)
 Conversely, if xa is a factor of P(x), then P(a) =0
6C  Factorisation of Polynomials
Remainder Theorem:
When P(x) is divided by bx+a, the remainder is P(a/b).
For example, if P(x) is divided by x1, let x1=0, x=1.
P(1) = Remainder (R(x))
For example, if P(x) is divided by 3x2, let 3x+2=0, x=2/3.
P(2/3) = Remainder (R(x))

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