Cheatography

# Chapter 6 Polynomials Cheat Sheet by liv.skreka

Cambridge Chapter 6 Polynomials Unit 1 - AOS 2- Polynomials

### 6A The Language of Polyno­mials

- A polynomial function is a function that can be written in the form:
P(x) = ax^n + ax^n-1 + ... + ax + a

- The leading term, ax^n, of a polynomial is the term of the highest index among those terms with a non-zero coeffi­cient.

### The function f(x)=x^1/3

Inverse functions
f(x)=x1/3 is the inverse of f(x)= x3

### 6E - Cubic Functions of f(x) = a(x-h)^3 + k

 Point of Inflection (POI) Vertical Transl­ations Horizontal Transl­ations A point of zero gradient by adding or subtra­cting a constant term to y-x^3, the graph moves either up or down The graph of y=(x-h)^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 x-axis.
The implied domain of all cubics is R and the range is also R

### 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.
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

Repeated roots/­factors

### 6F - Graphs of factorised cubic functions

Cubic equations with one x intercept

### 6H - Families of cubic polynomial functions

 y=ax^3 y=a(x-h)^3 + k y=a(x-­a)(­x-b­)(x-c) y=ax^3­+bx­^2+cx+d

### 6H - Families of cubic polynomial functions

Quartic Functions

### 6B - Division of Polyno­mials

Equating Coeffi­cients Methods instead of dividing

### 6C - Rational Root Theorem

 P(x) = 2x3 - x2 - 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, x-3/2, which equates to 2x-3 as a factor

### 6C - Factor­isation of Polyno­mials

For a polynomial P(x)
- If P(a) = 0, then x-a is a factor of P(x)
- Conver­sely, if x-a 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 x-1, let x-1=0, x=1.
P(1) = Remainder (R(x))

For example, if P(x) is divided by 3x-2, let 3x+2=0, x=-2/3.
P(-2/3) = Remainder (R(x))