Chapter 6 Polynomials Cheat Sheet by liv.skreka

- 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.

Inverse functions
f(x)=x^{1/3 is the inverse of f(x)= x}3

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.

- 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

Dividing Polyno­mials involving fractions

Dividing polyno­mials when we have a remainder

Equating Coeffi­cients Methods instead of dividing

Repeated roots/­factors

Cubic equations with one x intercept

Quartic Functions

Equating Coeffi­cients Methods instead of dividing

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

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))